%------------------------------------------------------------------------------
% File     : ITP286^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00347_018417
% 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_00347_018417 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12229 (5284 unt;1994 typ;   0 def)
%            Number of atoms       : 30577 (13273 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 139399 (3188   ~; 475   |;2104   &;121206   @)
%                                         (   0 <=>;12426  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   94 (   6 avg)
%            Number of types       :  267 ( 266 usr)
%            Number of type conns  : 8661 (8661   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1731 (1728 usr; 108 con; 0-8 aty)
%            Number of variables   : 30617 (3488   ^;26122   !;1007   ?;30617   :)
% 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 20:22:39.875
%------------------------------------------------------------------------------
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thf(ty_n_t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Heap____Time____Monad__OHeap_It__List__Olist_I_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__String__Ochar_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_It__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Set__Oset_It__String__Ochar_J_J,type,
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thf(ty_n_t__Heap__Oarray_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Real__Oreal_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(ty_n_t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(ty_n_t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
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thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
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thf(ty_n_t__Heap____Time____Monad__OHeap_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__List__Olist_It__Assertions__Oassn_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
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thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
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thf(ty_n_t__List__Olist_It__Uint32__Ouint32_J,type,
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thf(ty_n_t__Set__Oset_It__Uint32__Ouint32_J,type,
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thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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thf(ty_n_t__Heap__Oarray_It__Nat__Onat_J,type,
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thf(ty_n_t__Heap__Oarray_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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thf(ty_n_t__Rat__Orat,type,
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thf(ty_n_t__Num__Onum,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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% Explicit typings (1728)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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thf(sy_c_Array__Time_Ofreeze_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Array__Time_Olen_001_Eo,type,
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thf(sy_c_Array__Time_Olen_001t__Int__Oint,type,
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thf(sy_c_Array__Time_Olen_001t__Nat__Onat,type,
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thf(sy_c_Array__Time_Olen_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Array__Time_Onth_001_Eo,type,
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thf(sy_c_Array__Time_Onth_001t__Int__Oint,type,
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thf(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
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thf(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Array__Time_Onth_001t__Product____Type__Ounit,type,
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thf(sy_c_Array__Time_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Array__Time_Oof__list_001_Eo,type,
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thf(sy_c_Array__Time_Oof__list_001t__Int__Oint,type,
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thf(sy_c_Array__Time_Oof__list_001t__Nat__Onat,type,
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thf(sy_c_Array__Time_Oof__list_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Array__Time_Oupd_001_Eo,type,
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thf(sy_c_Array__Time_Oupd_001t__Int__Oint,type,
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thf(sy_c_Array__Time_Oupd_001t__Nat__Onat,type,
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thf(sy_c_Array__Time_Oupd_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Assertions_Oassn_ORep__assn,type,
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thf(sy_c_Assertions_Oentails,type,
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thf(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Assertions_Opure__assn,type,
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thf(sy_c_Assertions_Osnga__assn_001_Eo,type,
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thf(sy_c_Assertions_Osnga__assn_001t__Int__Oint,type,
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thf(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: array_nat > list_nat > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Option__Ooption_It__Nat__Onat_J,type,
    snga_assn_option_nat: array_option_nat > list_option_nat > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Product____Type__Ounit,type,
    snga_a4522542871529764173t_unit: array_Product_unit > list_Product_unit > 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__QUERY,type,
    fI_QUERY: 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_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).

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__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

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__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_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_Onegative,type,
    code_negative: num > code_integer ).

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

thf(sy_c_Code__Numeral_Opositive,type,
    code_positive: num > code_integer ).

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__Code____Numeral__Ointeger,type,
    code_T5410181022262596154nteger: ( code_integer > produc8923325533196201883nteger ) > code_integer > code_integer > code_integer ).

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_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Uint32__Ouint32,type,
    code_T8223577488154910793uint32: ( code_integer > produc827990862158126777uint32 ) > uint32 > code_integer > uint32 ).

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,
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thf(sy_c_Complex_Ocis,type,
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thf(sy_c_Complex_Ocomplex_OComplex,type,
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thf(sy_c_Complex_Ocsqrt,type,
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thf(sy_c_Complex_Oimaginary__unit,type,
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thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
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thf(sy_c_Divides_Oadjust__div,type,
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thf(sy_c_Divides_Oadjust__mod,type,
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thf(sy_c_Divides_Odivmod__nat,type,
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thf(sy_c_Divides_Oeucl__rel__int,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Uint32__Ouint32,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_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_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__String__Ochar_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_Heap__Time__Monad_Obind_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Heap__Time__Monad_Obind_001t__Nat__Onat_001t__Product____Type__Ounit,type,
    heap_T2412598413086283380t_unit: heap_Time_Heap_nat > ( nat > heap_T5738788834812785303t_unit ) > heap_T5738788834812785303t_unit ).

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__Product____Type__Ounit,type,
    heap_T2633723481742716231t_unit: heap_T5738788834812785303t_unit > ( product_unit > heap_T5738788834812785303t_unit ) > heap_T5738788834812785303t_unit ).

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__Nat__Onat,type,
    heap_T5387808048883414158Ti_nat: heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > heap_Time_Heap_nat ) > heap_Time_Heap_nat ).

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,
    heap_Time_effect_o: heap_Time_Heap_o > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o > nat > $o ).

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_Otap_001_Eo,type,
    heap_Time_tap_o: ( heap_e7401611519738050253t_unit > $o ) > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Otap_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T3993282329578657722on_nat: ( heap_e7401611519738050253t_unit > option_nat ) > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Otap_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T861569056895978319_VEBTi: ( heap_e7401611519738050253t_unit > 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__Heap__Oarray_I_Eo_J,type,
    hoare_6478655245392655262rray_o: assn > heap_T5660665574680485309rray_o > ( array_o > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Heap__Oarray_It__Int__Oint_J,type,
    hoare_2629421205684067388ay_int: assn > heap_T1346037964561226099ay_int > ( array_int > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Heap__Oarray_It__Nat__Onat_J,type,
    hoare_6807272225193264096ay_nat: assn > heap_T3836121109492952855ay_nat > ( array_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    hoare_3353465787467722821_VEBTi: assn > heap_T8822477325091257596_VEBTi > ( array_VEBT_VEBTi > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Int__Oint,type,
    hoare_3065115510600077593le_int: assn > heap_Time_Heap_int > ( int > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_I_Eo_J,type,
    hoare_9089481587091695345list_o: assn > heap_T844314716496656296list_o > ( list_o > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Nat__Onat_J,type,
    hoare_7964568885773372237st_nat: assn > heap_T290393402774840812st_nat > ( list_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    hoare_6480275734082232733on_nat: assn > heap_T5317711798761887292on_nat > ( list_option_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Product____Type__Ounit_J,type,
    hoare_8193881092815343078t_unit: assn > heap_T7268547540234007069t_unit > ( list_Product_unit > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    hoare_3904069481286416050_VEBTi: assn > heap_T4980287057938770641_VEBTi > ( list_VEBT_VEBTi > 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,
    hoare_7629718768684598413on_nat: assn > heap_T2636463487746394924on_nat > ( option_nat > assn ) > $o ).

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_Hoare__Triple_Onew__addrs,type,
    hoare_new_addrs: heap_e7401611519738050253t_unit > set_nat > heap_e7401611519738050253t_unit > set_nat ).

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

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    if_Heap_Time_Heap_o: $o > heap_Time_Heap_o > heap_Time_Heap_o > heap_Time_Heap_o ).

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    if_Hea811341299636385687_VEBTi: $o > heap_T4980287057938770641_VEBTi > heap_T4980287057938770641_VEBTi > heap_T4980287057938770641_VEBTi ).

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    if_Hea2662716070787841314ap_nat: $o > heap_Time_Heap_nat > heap_Time_Heap_nat > heap_Time_Heap_nat ).

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    if_Hea5867803462524415986on_nat: $o > heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
    if_Hea8138950348631371857t_unit: $o > heap_T5738788834812785303t_unit > heap_T5738788834812785303t_unit > heap_T5738788834812785303t_unit ).

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    if_Hea8453224502484754311_VEBTi: $o > heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi ).

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

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

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

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

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

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

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

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

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

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

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

thf(sy_c_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    if_Pro1135515155860407935uint32: $o > produc827990862158126777uint32 > produc827990862158126777uint32 > produc827990862158126777uint32 ).

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

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

thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(sy_c_If_001t__Uint32__Ouint32,type,
    if_uint32: $o > uint32 > uint32 > uint32 ).

thf(sy_c_If_001t__VEBT____BuildupMemImp__OVEBTi,type,
    if_VEBT_VEBTi: $o > vEBT_VEBTi > vEBT_VEBTi > vEBT_VEBTi ).

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

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,
    if_wor5778924947035936048l_num1: $o > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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

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

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

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

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

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

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

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

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

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Uint32__Ouint32,type,
    ring_1_of_int_uint32: int > uint32 ).

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,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
    least_4859182151741483524sb_int: int > $o ).

thf(sy_c_List_Oappend_001_Eo,type,
    append_o: list_o > list_o > list_o ).

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

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

thf(sy_c_List_Oenumerate_001_Eo,type,
    enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
    enumerate_int: nat > list_int > list_P3521021558325789923at_int ).

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

thf(sy_c_List_Oenumerate_001t__Real__Oreal,type,
    enumerate_real: nat > list_real > list_P3644420460460130531t_real ).

thf(sy_c_List_Oenumerate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    enumerate_VEBT_VEBTi: nat > list_VEBT_VEBTi > list_P2320588648998582380_VEBTi ).

thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
    enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
    filter_nat: ( nat > $o ) > list_nat > list_nat ).

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,
    foldr_assn_assn: ( assn > assn > assn ) > list_assn > assn > assn ).

thf(sy_c_List_Ofoldr_001t__Int__Oint_001t__Nat__Onat,type,
    foldr_int_nat: ( int > nat > nat ) > list_int > nat > nat ).

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,
    linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).

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

thf(sy_c_List_Olist_OCons_001_Eo,type,
    cons_o: $o > list_o > list_o ).

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,
    nil_o: list_o ).

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

thf(sy_c_List_Olist_Omap_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Olist_Omap_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
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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__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__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,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__Int__Oint,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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_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_001_Eo,type,
    map_nat_o: ( nat > $o ) > list_nat > list_o ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Oset_001_Eo,type,
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thf(sy_c_List_Olist_Oset_001t__Assertions__Oassn,type,
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thf(sy_c_List_Olist_Oset_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001_Eo,type,
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thf(sy_c_List_Olist__update_001t__Int__Oint,type,
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thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
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thf(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Onth_001_Eo,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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thf(sy_c_List_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Ounit,type,
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thf(sy_c_List_Onth_001t__Real__Oreal,type,
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thf(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
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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__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__Nat__Onat,type,
    produc2282297823089607884Ti_nat: list_VEBT_VEBTi > list_nat > list_P659468882601404396Ti_nat ).

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_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    replic6271474913745183777e_term: nat > ( int > option6357759511663192854e_term ) > list_i8448526496819171953e_term ).

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__Code____Numeral__Ointeger,type,
    replic7707675349574490269nteger: nat > code_integer > list_Code_integer ).

thf(sy_c_List_Oreplicate_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    replic5388364368018022029t_unit: nat > heap_e7401611519738050253t_unit > list_h2745205591237434579t_unit ).

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

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

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__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__Int__Oint_Mt__Int__Oint_J,type,
    replic1057375728873637753nt_int: nat > product_prod_int_int > list_P5707943133018811711nt_int ).

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

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

thf(sy_c_List_Oreplicate_001t__Uint32__Ouint32,type,
    replicate_uint32: nat > uint32 > list_uint32 ).

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_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    rel_of5543720577181062686nt_int: ( ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int ) > ( produc7773217078559923341nt_int > $o ) > set_Pr1872883991513573699nt_int ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    rel_of7835217753297429671nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > option2651255830984564193nteger ) > ( produc1908205239877642774nteger > $o ) > set_Pr1281608226676607948nteger ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    rel_of7774016450764239315et_nat: ( ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat ) > ( produc2732055786443039994et_nat > $o ) > set_Pr8536935166611901872et_nat ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    rel_of4838799251197538391et_nat: ( ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat ) > ( produc3925858234332021118et_nat > $o ) > set_Pr3286484037609594932et_nat ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    rel_of8306664904814525588nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > option4624381673175914239nt_int ) > ( produc2285326912895808259nt_int > $o ) > set_Pr9222295170931077689nt_int ).

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_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__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_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Ointeger,type,
    semiri4055485073559036834nteger: ( code_integer > code_integer ) > nat > code_integer > code_integer ).

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

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

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Uint32__Ouint32,type,
    semiri2064589214733661617uint32: ( uint32 > uint32 ) > nat > uint32 > uint32 ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_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,
    semiri2846968517960172219l_num1: ( word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ) > nat > word_N3645301735248828278l_num1 > 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_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    size_s8881356780637927685e_term: list_i8448526496819171953e_term > 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__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J,type,
    size_s7011499401410611007t_unit: list_h2745205591237434579t_unit > 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__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6086282163384603972on_nat: list_option_nat > nat ).

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__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__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__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__Product____Type__Ounit_J,type,
    size_s245203480648594047t_unit: list_Product_unit > nat ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Uint32__Ouint32_J,type,
    size_s4844771616002835472uint32: list_uint32 > 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__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Uint32__Ouint32,type,
    neg_nu4269007558841261821uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_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_nu8115118780965096967l_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_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P3773570700014501484nt_int: option4256020574406277085nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    none_P4442379456014020469nteger: option8051342751916580710nteger ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    none_P199884684680593241et_nat: option2860828798490689354et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    none_P4972525538344268765et_nat: option5190343406534369742et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P1286213070022356066nt_int: option7541221861074943443nt_int ).

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__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__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__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__Assertions__Oassn,type,
    some_assn: assn > option_assn ).

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_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P7455497367792166888nt_int: produc7773217078559923341nt_int > option4256020574406277085nt_int ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    some_P1462369734362851057nteger: produc1908205239877642774nteger > option8051342751916580710nteger ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    some_P1630309045189364437et_nat: produc2732055786443039994et_nat > option2860828798490689354et_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    some_P750831030444334937et_nat: produc3925858234332021118et_nat > option5190343406534369742et_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P2355398578364412894nt_int: produc2285326912895808259nt_int > option7541221861074943443nt_int ).

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__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    some_P6772290148444788224nteger: produc8923325533196201883nteger > option2651255830984564193nteger ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    some_P624177172695371229et_nat: produc3658429121746597890et_nat > option936205604648967762et_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__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__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__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,
    some_P696572436114257607it_nat: produc2451868925425708898it_nat > option5408194888911472936it_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__Int__Oint_J,type,
    some_set_int: set_int > option_set_int ).

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_OSome_001t__Set__Oset_It__Real__Oreal_J,type,
    some_set_real: set_real > option_set_real ).

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__String__Ochar_J,type,
    some_set_char: set_char > option_set_char ).

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__Int__Oint_J,type,
    case_o223999843215110191et_int: $o > ( set_int > $o ) > option_set_int > $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__Product____Type__Ounit_J_001t__Nat__Onat,type,
    case_o8044209130310455284it_nat: heap_T5738788834812785303t_unit > ( nat > heap_T5738788834812785303t_unit ) > option_nat > heap_T5738788834812785303t_unit ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_001t__Num__Onum,type,
    case_o4601540916466233918it_num: heap_T5738788834812785303t_unit > ( num > heap_T5738788834812785303t_unit ) > option_num > heap_T5738788834812785303t_unit ).

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_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__String__Ochar_J,type,
    bot_bot_set_char: set_char ).

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_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__Int__Oint_J_J,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le2124322318746777828_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    ord_le4340812435750786203eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J_J,type,
    ord_le6753239538765779593_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    ord_le8000401564054156549_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le5643404153117327598_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_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_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_062_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J_J,type,
    ord_le2556027599737686990_num_o: ( product_prod_num_num > product_prod_num_num > $o ) > ( product_prod_num_num > product_prod_num_num > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_M_Eo_J,type,
    ord_le2239182809043710856_num_o: ( product_prod_num_num > $o ) > ( product_prod_num_num > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le1077754993875142464_nat_o: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le7812727212727832188_nat_o: ( produc9072475918466114483BT_nat > $o ) > ( produc9072475918466114483BT_nat > $o ) > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le418104280809901481VEBT_o: ( vEBT_VEBT > $o ) > ( vEBT_VEBT > $o ) > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__Int__Oint_J_J,type,
    ord_le353528952715127954et_int: option_set_int > option_set_int > $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_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
    ord_le4763372923235995152et_nat: set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    ord_le5966269811547037844et_nat: set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $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__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__String__Ochar_J,type,
    ord_less_eq_set_char: set_char > set_char > $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____BuildupMemImp__OVEBTi_J,type,
    ord_le6592769550269828683_VEBTi: set_VEBT_VEBTi > set_VEBT_VEBTi > $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__Int__Oint,type,
    ord_max_int: int > int > int ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__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__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_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    top_to6513043852502318900_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    top_to8112647782992986859eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J_J,type,
    top_to7190503160269336793_nat_o: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    top_to2428096842796733269_nat_o: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    top_to6261655714344447806_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Assertions__Oassn_J,type,
    top_top_option_assn: option_assn ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    top_to4826455019444611206et_nat: option_set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Real__Oreal_J_J,type,
    top_to1083748111577038690t_real: option_set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__String__Ochar_J_J,type,
    top_to6745749650031393671t_char: option_set_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    top_to4645266643341252675nteger: set_Code_integer ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    top_top_set_complex: set_complex ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

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__Option__Ooption_It__Code____Numeral__Ointeger_J_J,type,
    top_to5929521628599800467nteger: set_op687863988967635939nteger ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Complex__Ocomplex_J_J,type,
    top_to6180147692022559204omplex: set_option_complex ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
    top_to6430115241214627170on_int: set_option_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    top_to8920198386146353926on_nat: set_option_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Real__Oreal_J_J,type,
    top_to853713521313446370n_real: set_option_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__String__Ochar_J_J,type,
    top_to576892095687177735n_char: set_option_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    top_to2069137843433766899nt_int: set_Pr1872883991513573699nt_int ).

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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,
    produc275000359906850836_nat_o: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o ) > produc3625547720036274456_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o ).

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,
    produc2677327216024927295_nat_o: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > vEBT_VEBTi > nat > $o ) > produc3625547720036274456_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > vEBT_VEBTi > nat > $o ).

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,
    produc2327743382103342416Heap_o: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o ) > produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o ).

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,
    produc1061038227461121684on_nat: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ) > produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat ).

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,
    produc2298712477539903273_VEBTi: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ) > produc3625547720036274456_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc6230002227079971283nt_int: produc7773217078559923341nt_int > int > option6357759511663192854e_term ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc7822682618958472924nteger: produc1908205239877642774nteger > produc6241069584506657477e_term > option6357759511663192854e_term ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc180342877477747464et_nat: produc2732055786443039994et_nat > produc3658429121746597890et_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc995936583742144908et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc6816164490631068361nt_int: produc2285326912895808259nt_int > produc8551481072490612790e_term > option6357759511663192854e_term ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    produc9167289414957590229n_assn: produc6575502325842934193n_assn > assn ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc1824681642469235216et_nat: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001_Eo,type,
    product_fst_nat_o: product_prod_nat_o > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Int__Oint,type,
    product_fst_nat_int: product_prod_nat_int > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Real__Oreal,type,
    product_fst_nat_real: produc7716430852924023517t_real > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc8252055991070844170_VEBTi: produc214224863196444774_VEBTi > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc8575180428842422559T_VEBT: produc8025551001238799321T_VEBT > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    produc9004433772639906525uint32: produc827990862158126777uint32 > uint32 ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc4993121158135996263VEBT_o: produc334124729049499915VEBT_o > vEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc8711427728657393693BT_int: produc4894624898956917775BT_int > vEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc8713918199166443969BT_nat: produc9072475918466114483BT_nat > vEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8110914911036349469T_real: produc5170161368751668367T_real > vEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc3162348030201620241nt_int: produc7773217078559923341nt_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc7856867400915047194nteger: produc1908205239877642774nteger > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc5374455773327741254et_nat: produc2732055786443039994et_nat > produc3925858234332021118et_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc4011572625026189258et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc7328097813583171335nt_int: produc2285326912895808259nt_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Osnd_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    produc7938581201502569057T_VEBT: produc2504756804600209347T_VEBT > vEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    produc2051961928117032727n_assn: produc6575502325842934193n_assn > assn ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc8586169260539613262et_nat: produc3658429121746597890et_nat > set_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc1678900780639429121T_VEBT: produc1531783533982839933T_VEBT > vEBT_VEBT ).

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__Real__Oreal,type,
    product_snd_nat_real: produc7716430852924023517t_real > real ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc271786961351840588_VEBTi: produc214224863196444774_VEBTi > vEBT_VEBTi ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc8172668247895388509T_VEBT: produc8025551001238799321T_VEBT > vEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc5083336317046741121T_VEBT: produc3757001726724277373T_VEBT > vEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    produc1510406741064981791uint32: produc827990862158126777uint32 > uint32 ).

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,
    type_N8448461349408098053l_num1: itself8794530163899892676l_num1 ).

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_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > complex > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: 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__Product____Type__Ounit,type,
    refine451278484176762712t_unit: heap_T5738788834812785303t_unit > heap_T5738788834812785303t_unit > $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 ).

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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,
    signed6753297604338940182l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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,
    ascii_of: char > char ).

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

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

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

thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
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thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_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_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax2440797749228149777t_unit: ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_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_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax2089144231571168657t_unit: ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_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_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax4084839521278773041t_unit: ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_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_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax3516980881688143089t_unit: ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_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_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    syntax5181832503971434150t_unit: ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    syntax7398250324933576852n_assn: assn > assn > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001_Eo,type,
    time_TBOUND_o: heap_Time_Heap_o > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_I_Eo_J,type,
    time_TBOUND_array_o: heap_T5660665574680485309rray_o > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Int__Oint_J,type,
    time_T8667231999468956601ay_int: heap_T1346037964561226099ay_int > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Nat__Onat_J,type,
    time_T3621710982123377501ay_nat: heap_T3836121109492952855ay_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T6070283812100419266_VEBTi: heap_T8822477325091257596_VEBTi > nat > $o ).

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__Product____Type__Ounit,type,
    time_h7375078312994015913t_unit: assn > heap_T5738788834812785303t_unit > ( product_unit > 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_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_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__Real__Oreal,type,
    cos_real: real > real ).

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

thf(sy_c_Transcendental_Otanh_001t__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__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_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__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_001_Eo_001_Eo,type,
    vEBT_L7363604446928714179sn_o_o: ( $o > $o > assn ) > list_o > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Int__Oint,type,
    vEBT_L4782520652837395649_o_int: ( $o > int > assn ) > list_o > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L4785011123346445925_o_nat: ( $o > nat > assn ) > list_o > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Real__Oreal,type,
    vEBT_L4725278957065240257o_real: ( $o > real > assn ) > list_o > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001_Eo,type,
    vEBT_L6066640139021943271_int_o: ( int > $o > assn ) > list_int > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( int > real > assn ) > list_int > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001_Eo,type,
    vEBT_L7887682484454631235_nat_o: ( nat > $o > assn ) > list_nat > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Int__Oint,type,
    vEBT_L8298612041380073281at_int: ( nat > int > assn ) > list_nat > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Nat__Onat,type,
    vEBT_L8301102511889123557at_nat: ( nat > nat > assn ) > list_nat > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( nat > real > assn ) > list_nat > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8158188754432654943T_VEBT: ( nat > vEBT_VEBT > assn ) > list_nat > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L6234343332106409831real_o: ( real > $o > assn ) > list_real > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Int__Oint,type,
    vEBT_L1443519841834266653al_int: ( real > int > assn ) > list_real > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( real > nat > assn ) > list_real > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( real > real > assn ) > list_real > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( real > vEBT_VEBT > assn ) > list_real > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7489408758114837031VEBT_o: ( vEBT_VEBT > $o > assn ) > list_VEBT_VEBT > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8294436054247626077BT_int: ( vEBT_VEBT > int > assn ) > list_VEBT_VEBT > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( vEBT_VEBT > nat > assn ) > list_VEBT_VEBT > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L8010285020845282001on_nat: ( vEBT_VEBT > option_nat > assn ) > list_VEBT_VEBT > list_option_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Ounit,type,
    vEBT_L8068554427805421084t_unit: ( vEBT_VEBT > product_unit > assn ) > list_VEBT_VEBT > list_Product_unit > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( vEBT_VEBT > real > assn ) > list_VEBT_VEBT > list_real > 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__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( vEBT_VEBT > vEBT_VEBT > assn ) > list_VEBT_VEBT > list_VEBT_VEBT > 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_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
    vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_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_V482137685244371085it_nat: produc5059602919146741221it_nat > produc5059602919146741221it_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_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_V3802522976469930835it_nat: produc6872358179685758443it_nat > produc6872358179685758443it_nat > $o ).

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_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_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_Mt__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_Mt__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_J,type,
    accp_P4085165796030815644it_nat: ( produc5059602919146741221it_nat > produc5059602919146741221it_nat > $o ) > produc5059602919146741221it_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_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_Mt__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_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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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,
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thf(sy_c_Wellfounded_Ofinite__psubset_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
    finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

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

thf(sy_c_member_001_Eo,type,
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thf(sy_c_member_001t__Assertions__Oassn,type,
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thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_member_001t__Complex__Ocomplex,type,
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thf(sy_c_member_001t__Int__Oint,type,
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thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_member_001t__Nat__Onat,type,
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thf(sy_c_member_001t__Num__Onum,type,
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    member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member6260224972018164377et_nat: produc3658429121746597890et_nat > set_Pr3948176798113811640et_nat > $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__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__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    member4307123515891402160nteger: produc6491284506569428743nteger > set_Pr7577011563204128103nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
    member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    member8027108493173000802uint32: produc827990862158126777uint32 > set_Pr1773385645901665561uint32 > $o ).

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

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

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

thf(sy_c_member_001t__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_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_tree__is,type,
    tree_is: list_VEBT_VEBTi ).

thf(sy_v_va____,type,
    va: nat ).

thf(sy_v_x11______,type,
    x11: option4927543243414619207at_nat ).

thf(sy_v_x12______,type,
    x12: nat ).

thf(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

thf(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10178)
thf(fact_0_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_1_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_2_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_3_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_4_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_5_mimapr,axiom,
    ord_less_eq_nat @ mi @ ma ).

% mimapr
thf(fact_6_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% high_def
thf(fact_7_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_mint @ T ) ) ) ) ) ).

% vebt_minti_h
thf(fact_8_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_maxt @ T ) ) ) ) ) ).

% vebt_maxti_h
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__C7_OIH_C_I2_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: option_nat,N2: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ ma @ xa )
     => ( ( Xa
          = ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
       => ( ( Xb
            = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( ord_less_nat @ Xb @ ( size_s6755466524823107622T_VEBT @ treeList ) )
           => ( ( Xc
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ Xb ) ) )
             => ( ~ ( ( Xc != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ Xa ) @ Xc ) )
               => ( ( vEBT_invar_vebt @ summary @ N2 )
                 => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ summary @ Ti ) @ ( vEBT_VEBT_vebt_predi @ summary @ Ti @ Xb )
                    @ ^ [R: option_nat] :
                        ( times_times_assn @ ( vEBT_vebt_assn_raw @ summary @ Ti )
                        @ ( pure_assn
                          @ ( R
                            = ( vEBT_vebt_pred @ summary @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(2)
thf(fact_11__C7_OIH_C_I1_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: option_nat,N2: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ ma @ xa )
     => ( ( Xa
          = ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
       => ( ( Xb
            = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( ord_less_nat @ Xb @ ( size_s6755466524823107622T_VEBT @ treeList ) )
           => ( ( Xc
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ Xb ) ) )
             => ( ( ( Xc != none_nat )
                  & ( vEBT_VEBT_greater @ ( some_nat @ Xa ) @ Xc ) )
               => ( ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ Xb ) @ N2 )
                 => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ treeList @ Xb ) @ Ti ) @ ( vEBT_VEBT_vebt_predi @ ( nth_VEBT_VEBT @ treeList @ Xb ) @ Ti @ Xa )
                    @ ^ [R: option_nat] :
                        ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ treeList @ Xb ) @ Ti )
                        @ ( pure_assn
                          @ ( R
                            = ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ treeList @ Xb ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(1)
thf(fact_12_pred__less__length__list,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_13_pred__lesseq__max,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_14_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_maxt @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_hT
thf(fact_15_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_mint @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_hT
thf(fact_16_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_17_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_18_power__increasing__iff,axiom,
    ! [B: real,X2: nat,Y2: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_19_power__increasing__iff,axiom,
    ! [B: rat,X2: nat,Y2: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_20_power__increasing__iff,axiom,
    ! [B: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_21_power__increasing__iff,axiom,
    ! [B: int,X2: nat,Y2: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_22_power__strict__increasing__iff,axiom,
    ! [B: real,X2: nat,Y2: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_23_power__strict__increasing__iff,axiom,
    ! [B: rat,X2: nat,Y2: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_24_power__strict__increasing__iff,axiom,
    ! [B: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_25_power__strict__increasing__iff,axiom,
    ! [B: int,X2: nat,Y2: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_26_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_27_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_28_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_29_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_30_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N2 )
     => ( Deg = N2 ) ) ).

% deg_deg_n
thf(fact_31_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_32_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList2 @ S ) ) ) ).

% deg_SUcn_Node
thf(fact_33_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_34_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_35_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_36_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_num__double,axiom,
    ! [N2: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N2 )
      = ( bit0 @ N2 ) ) ).

% num_double
thf(fact_56_misiz,axiom,
    ! [T: vEBT_VEBT,N2: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% misiz
thf(fact_57_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ one_one_rat @ N2 )
      = one_one_rat ) ).

% power_one
thf(fact_58_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_uint32 @ one_one_uint32 @ N2 )
      = one_one_uint32 ) ).

% power_one
thf(fact_59_power__one,axiom,
    ! [N2: nat] :
      ( ( power_2184487114949457152l_num1 @ one_on7727431528512463931l_num1 @ N2 )
      = one_on7727431528512463931l_num1 ) ).

% power_one
thf(fact_60_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ one_one_real @ N2 )
      = one_one_real ) ).

% power_one
thf(fact_61_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ one_one_nat @ N2 )
      = one_one_nat ) ).

% power_one
thf(fact_62_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ one_one_int @ N2 )
      = one_one_int ) ).

% power_one
thf(fact_63_power__mult__numeral,axiom,
    ! [A: uint32,M: num,N2: num] :
      ( ( power_power_uint32 @ ( power_power_uint32 @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_uint32 @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_64_power__mult__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,M: num,N2: num] :
      ( ( power_2184487114949457152l_num1 @ ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_65_power__mult__numeral,axiom,
    ! [A: real,M: num,N2: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_66_power__mult__numeral,axiom,
    ! [A: nat,M: num,N2: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_67_power__mult__numeral,axiom,
    ! [A: int,M: num,N2: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_68_helpypredd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Y2 ) )
       => ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpypredd
thf(fact_69_power__one__right,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_70_power__one__right,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_71_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_72_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_73_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_74_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_75_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_76_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_77_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_78_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_79_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_80_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_81_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_real @ N2 )
        = one_one_real )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_82_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_rat @ N2 )
        = one_one_rat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_83_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_84_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_int @ N2 )
        = one_one_int )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_85_power__inject__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_86_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_87_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_88_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_89_pred__list__to__short,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_90_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_91_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_92_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_93_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_94_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
    ! [A2: set_complex] :
      ( ( collect_complex
        @ ^ [X: complex] : ( member_complex @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_115_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_116_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_117_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_118_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_119__C7_Oprems_C,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ na ).

% "7.prems"
thf(fact_120_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_121_le__num__One__iff,axiom,
    ! [X2: num] :
      ( ( ord_less_eq_num @ X2 @ one )
      = ( X2 = one ) ) ).

% le_num_One_iff
thf(fact_122_power__mult,axiom,
    ! [A: uint32,M: nat,N2: nat] :
      ( ( power_power_uint32 @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_uint32 @ ( power_power_uint32 @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_123_power__mult,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_2184487114949457152l_num1 @ ( power_2184487114949457152l_num1 @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_124_power__mult,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_125_power__mult,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_126_power__mult,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_127_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_128_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_129_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_130_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_131_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_132_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_133_power__even__eq,axiom,
    ! [A: uint32,N2: nat] :
      ( ( power_power_uint32 @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_uint32 @ ( power_power_uint32 @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_134_power__even__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_2184487114949457152l_num1 @ ( power_2184487114949457152l_num1 @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_135_power__even__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_136_power__even__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_137_power__even__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_138_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_139_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_140_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_141_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_142_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_143_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_144_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_145_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_146_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_147_power__commuting__commutes,axiom,
    ! [X2: uint32,Y2: uint32,N2: nat] :
      ( ( ( times_times_uint32 @ X2 @ Y2 )
        = ( times_times_uint32 @ Y2 @ X2 ) )
     => ( ( times_times_uint32 @ ( power_power_uint32 @ X2 @ N2 ) @ Y2 )
        = ( times_times_uint32 @ Y2 @ ( power_power_uint32 @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_148_power__commuting__commutes,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,N2: nat] :
      ( ( ( times_7065122842183080059l_num1 @ X2 @ Y2 )
        = ( times_7065122842183080059l_num1 @ Y2 @ X2 ) )
     => ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) @ Y2 )
        = ( times_7065122842183080059l_num1 @ Y2 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_149_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_150_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_151_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_152_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_153_power__mult__distrib,axiom,
    ! [A: uint32,B: uint32,N2: nat] :
      ( ( power_power_uint32 @ ( times_times_uint32 @ A @ B ) @ N2 )
      = ( times_times_uint32 @ ( power_power_uint32 @ A @ N2 ) @ ( power_power_uint32 @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_154_power__mult__distrib,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ ( times_7065122842183080059l_num1 @ A @ B ) @ N2 )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ N2 ) @ ( power_2184487114949457152l_num1 @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_155_power__mult__distrib,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_156_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_157_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_158_power__mult__distrib,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_159_power__commutes,axiom,
    ! [A: uint32,N2: nat] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ A @ N2 ) @ A )
      = ( times_times_uint32 @ A @ ( power_power_uint32 @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_160_power__commutes,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ N2 ) @ A )
      = ( times_7065122842183080059l_num1 @ A @ ( power_2184487114949457152l_num1 @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_161_power__commutes,axiom,
    ! [A: real,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_162_power__commutes,axiom,
    ! [A: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_163_power__commutes,axiom,
    ! [A: nat,N2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_164_power__commutes,axiom,
    ! [A: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_165_power__divide,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
      = ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).

% power_divide
thf(fact_166_power__divide,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N2 )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).

% power_divide
thf(fact_167_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_168_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_169_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_170_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_171_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_172_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_173_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_174_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_175_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_176_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_177_numeral__One,axiom,
    ( ( numera7442385471795722001l_num1 @ one )
    = one_on7727431528512463931l_num1 ) ).

% numeral_One
thf(fact_178_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_179_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_180_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_181_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_182_mult__numeral__1__right,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A @ ( numera7442385471795722001l_num1 @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_183_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_184_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_185_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_186_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_187_mult__numeral__1,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_188_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_189_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_190_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_191_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_192_one__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_193_one__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_194_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_195_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_196_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_197_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_198_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_199_left__right__inverse__power,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,N2: nat] :
      ( ( ( times_7065122842183080059l_num1 @ X2 @ Y2 )
        = one_on7727431528512463931l_num1 )
     => ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) @ ( power_2184487114949457152l_num1 @ Y2 @ N2 ) )
        = one_on7727431528512463931l_num1 ) ) ).

% left_right_inverse_power
thf(fact_200_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_201_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_202_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_203_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_204_power__Suc2,axiom,
    ! [A: uint32,N2: nat] :
      ( ( power_power_uint32 @ A @ ( suc @ N2 ) )
      = ( times_times_uint32 @ ( power_power_uint32 @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_205_power__Suc2,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ A @ ( suc @ N2 ) )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_206_power__Suc2,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_207_power__Suc2,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_208_power__Suc2,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_209_power__Suc2,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_210_power__Suc,axiom,
    ! [A: uint32,N2: nat] :
      ( ( power_power_uint32 @ A @ ( suc @ N2 ) )
      = ( times_times_uint32 @ A @ ( power_power_uint32 @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_211_power__Suc,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ A @ ( suc @ N2 ) )
      = ( times_7065122842183080059l_num1 @ A @ ( power_2184487114949457152l_num1 @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_212_power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_213_power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_214_power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_215_power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_216_power__one__over,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_217_power__one__over,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_218_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_219_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_220_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_221_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_222_power__less__power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_223_power__less__power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_224_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_225_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_226_power__gt1__lemma,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_227_power__gt1__lemma,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_228_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_229_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_230_power__gt1,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_231_power__gt1,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_232_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_233_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_234_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_235_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_236_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_237_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_238_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_239_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_240_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_241_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_242_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_243_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_244_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_245_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_246_power__numeral__even,axiom,
    ! [Z: uint32,W: num] :
      ( ( power_power_uint32 @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_uint32 @ ( power_power_uint32 @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_uint32 @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_247_power__numeral__even,axiom,
    ! [Z: word_N3645301735248828278l_num1,W: num] :
      ( ( power_2184487114949457152l_num1 @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_2184487114949457152l_num1 @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_248_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_249_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_250_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_251_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_252_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_253_one__power2,axiom,
    ( ( power_power_uint32 @ one_one_uint32 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% one_power2
thf(fact_254_one__power2,axiom,
    ( ( power_2184487114949457152l_num1 @ one_on7727431528512463931l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_on7727431528512463931l_num1 ) ).

% one_power2
thf(fact_255_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_256_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_257_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_258_power2__eq__square,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_uint32 @ A @ A ) ) ).

% power2_eq_square
thf(fact_259_power2__eq__square,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_7065122842183080059l_num1 @ A @ A ) ) ).

% power2_eq_square
thf(fact_260_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_261_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_262_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_263_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_264_power4__eq__xxxx,axiom,
    ! [X2: uint32] :
      ( ( power_power_uint32 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_uint32 @ ( times_times_uint32 @ ( times_times_uint32 @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_265_power4__eq__xxxx,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_266_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_267_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_268_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_269_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_270_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_271_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_272_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_273_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_274_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_275_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_276_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_277_power__odd__eq,axiom,
    ! [A: uint32,N2: nat] :
      ( ( power_power_uint32 @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_uint32 @ A @ ( power_power_uint32 @ ( power_power_uint32 @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_278_power__odd__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_7065122842183080059l_num1 @ A @ ( power_2184487114949457152l_num1 @ ( power_2184487114949457152l_num1 @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_279_power__odd__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_280_power__odd__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_281_power__odd__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_282_power__odd__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_283_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_284_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X2 ) @ one_one_nat ) ) ).

% tdeletemimi'
thf(fact_285__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__predi_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060_092_060lambda_062r_O_Avebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_A_K_A_092_060up_062_A_Ir_A_061_Avebt__pred_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: $o,X22: $o] :
      ( ( tia
        = ( vEBT_Leafi @ X21 @ X22 ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia ) @ ( vEBT_VEBT_vebt_predi @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia @ xa )
        @ ^ [R: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ xa ) ) ) ) ) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_predi' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <\<lambda>r. vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti * \<up> (r = vebt_pred (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
thf(fact_286_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X2 ) ) ) ) ).

% maxt_sound
thf(fact_287_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).

% maxt_corr
thf(fact_288_mint__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X2 ) ) ) ) ).

% mint_sound
thf(fact_289_mint__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).

% mint_corr
thf(fact_290_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( N2
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_291_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X2 = Mi )
          | ( X2 = Ma )
          | ( ( ord_less_nat @ X2 @ Ma )
            & ( ord_less_nat @ Mi @ X2 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_292_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_293_lowi__hT,axiom,
    ! [X2: nat,N2: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_low @ X2 @ N2 ) ) )
      @ one_one_nat ) ).

% lowi_hT
thf(fact_294_highi__hT,axiom,
    ! [X2: nat,N2: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_high @ X2 @ N2 ) ) )
      @ one_one_nat ) ).

% highi_hT
thf(fact_295_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_296_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_297_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_298_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_299_mint__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_300_maxt__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_301_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Mini: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X2 )
         => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).

% mint_corr_help
thf(fact_302_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X2 )
         => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_303_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_304_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_305_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_306_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_eq_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_307_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_308_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_309_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_310_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_311_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_312_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_313_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_314_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_315_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_316_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_317_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_318_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_319_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_320_helpyd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Y2 ) )
       => ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpyd
thf(fact_321_succ__min,axiom,
    ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_322_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ T @ X2 )
        = ( member_nat @ X2 @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_323_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_324_distrib__left__numeral,axiom,
    ! [V: num,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( plus_p361126936061061375l_num1 @ B @ C ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ B ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_325_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_326_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_327_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_328_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_329_distrib__right__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,V: num] :
      ( ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ A @ B ) @ ( numera7442385471795722001l_num1 @ V ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ A @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ B @ ( numera7442385471795722001l_num1 @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_330_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_331_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_332_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_333_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_one__add__one,axiom,
    ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ one_on7727431528512463931l_num1 )
    = ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_345_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_346_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_347_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_348_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_349_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_350_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_351_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_352_pred__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T @ Y2 )
        & ( ord_less_nat @ Y2 @ X2 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ Z2 @ X2 ) )
           => ( ord_less_eq_nat @ Z2 @ Y2 ) ) ) ) ).

% pred_member
thf(fact_353_succ__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T @ Y2 )
        & ( ord_less_nat @ X2 @ Y2 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ X2 @ Z2 ) )
           => ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ) ).

% succ_member
thf(fact_354_pred__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Px ) ) ) ).

% pred_corr
thf(fact_355_succ__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% succ_corr
thf(fact_356_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X2 )
        @ ^ [R: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_succ @ T @ X2 ) ) ) ) ) ) ).

% vebt_succi'_rf_abstr
thf(fact_357_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_358_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_359_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_360_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit0 @ N2 ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) @ ( numera7442385471795722001l_num1 @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_power__add,axiom,
    ! [A: uint32,M: nat,N2: nat] :
      ( ( power_power_uint32 @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_uint32 @ ( power_power_uint32 @ A @ M ) @ ( power_power_uint32 @ A @ N2 ) ) ) ).

% power_add
thf(fact_371_power__add,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ M ) @ ( power_2184487114949457152l_num1 @ A @ N2 ) ) ) ).

% power_add
thf(fact_372_power__add,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_add
thf(fact_373_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_add
thf(fact_374_power__add,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_375_power__add,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_376_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_377_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_378_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_379_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_380_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_381_mult__2,axiom,
    ! [Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ Z )
      = ( plus_p361126936061061375l_num1 @ Z @ Z ) ) ).

% mult_2
thf(fact_382_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_383_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_384_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_385_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_386_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_387_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_388_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_389_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_390_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_391_left__add__twice,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A @ ( plus_p361126936061061375l_num1 @ A @ B ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_392_left__add__twice,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_393_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_394_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_395_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_396_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_397_power2__sum,axiom,
    ! [X2: uint32,Y2: uint32] :
      ( ( power_power_uint32 @ ( plus_plus_uint32 @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_uint32 @ ( plus_plus_uint32 @ ( power_power_uint32 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_uint32 @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_398_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_399_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_400_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_401_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_402_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_403_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N4: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_404_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N4: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_405_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_406_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_407_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( ( X2 != Mi )
       => ( ( X2 != Ma )
         => ( ~ ( ord_less_nat @ X2 @ Mi )
            & ( ~ ( ord_less_nat @ X2 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X2 )
                & ( ~ ( ord_less_nat @ Ma @ X2 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_408_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X2 ) @ Y2 )
           => ( ( vEBT_vebt_member @ T @ Y2 )
              | ( X2 = Y2 ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_409_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [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] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
      = ( 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 ) ) ) ) ) ).

% vebt_assn_raw.simps(2)
thf(fact_410_insert__simp__mima,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        | ( X2 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_411_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M2: nat] :
              ( ( ( some_nat @ M2 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_412_nth__rule,axiom,
    ! [I: nat,Xs2: list_int,A: array_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( hoare_3065115510600077593le_int @ ( snga_assn_int @ A @ Xs2 ) @ ( array_nth_int @ A @ I )
        @ ^ [R: int] :
            ( times_times_assn @ ( snga_assn_int @ A @ Xs2 )
            @ ( pure_assn
              @ ( R
                = ( nth_int @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_413_nth__rule,axiom,
    ! [I: nat,Xs2: list_option_nat,A: array_option_nat] :
      ( ( ord_less_nat @ I @ ( size_s6086282163384603972on_nat @ Xs2 ) )
     => ( hoare_7629718768684598413on_nat @ ( snga_assn_option_nat @ A @ Xs2 ) @ ( array_nth_option_nat @ A @ I )
        @ ^ [R: option_nat] :
            ( times_times_assn @ ( snga_assn_option_nat @ A @ Xs2 )
            @ ( pure_assn
              @ ( R
                = ( nth_option_nat @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_414_nth__rule,axiom,
    ! [I: nat,Xs2: list_nat,A: array_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A @ Xs2 ) @ ( array_nth_nat @ A @ I )
        @ ^ [R: nat] :
            ( times_times_assn @ ( snga_assn_nat @ A @ Xs2 )
            @ ( pure_assn
              @ ( R
                = ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_415_nth__rule,axiom,
    ! [I: nat,Xs2: list_o,A: array_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( hoare_hoare_triple_o @ ( snga_assn_o @ A @ Xs2 ) @ ( array_nth_o @ A @ I )
        @ ^ [R: $o] :
            ( times_times_assn @ ( snga_assn_o @ A @ Xs2 )
            @ ( pure_assn
              @ ( R
                = ( nth_o @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_416_nth__rule,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,A: array_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( hoare_1429296392585015714_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_nth_VEBT_VEBTi @ A @ I )
        @ ^ [R: vEBT_VEBTi] :
            ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 )
            @ ( pure_assn
              @ ( R
                = ( nth_VEBT_VEBTi @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_417_nth__rule,axiom,
    ! [I: nat,Xs2: list_Product_unit,A: array_Product_unit] :
      ( ( ord_less_nat @ I @ ( size_s245203480648594047t_unit @ Xs2 ) )
     => ( hoare_8945653483474564448t_unit @ ( snga_a4522542871529764173t_unit @ A @ Xs2 ) @ ( array_7872002506669749220t_unit @ A @ I )
        @ ^ [R: product_unit] :
            ( times_times_assn @ ( snga_a4522542871529764173t_unit @ A @ Xs2 )
            @ ( pure_assn
              @ ( R
                = ( nth_Product_unit @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_418_power__2__mult__step__le,axiom,
    ! [N5: nat,N2: nat,K2: nat,K: nat] :
      ( ( ord_less_eq_nat @ N5 @ N2 )
     => ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ K2 ) @ ( 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 ) ) @ N5 ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_419_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_420_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_421_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_422_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_423_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ( hoare_8945653483474564448t_unit @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_8945653483474564448t_unit @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_424_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_425_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_426_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_427_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_428_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_429_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ( hoare_8945653483474564448t_unit @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_8945653483474564448t_unit @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_430_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% less_shift
thf(fact_431_maxbmo,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X2 ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X2 ) ) ).

% maxbmo
thf(fact_432_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
        = ( vEBT_vebt_member @ T @ X2 ) ) ) ).

% both_member_options_equiv_member
thf(fact_433_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
       => ( vEBT_vebt_member @ T @ X2 ) ) ) ).

% valid_member_both_member_options
thf(fact_434_highi__h,axiom,
    ! [X2: nat,N2: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_high @ X2 @ N2 ) ) ) ) ).

% highi_h
thf(fact_435_lowi__h,axiom,
    ! [X2: nat,N2: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).

% lowi_h
thf(fact_436_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Nodei @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
thf(fact_437_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leafi @ X21 @ X22 )
        = ( vEBT_Leafi @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% VEBTi.inject(2)
thf(fact_438_pred__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% pred_correct
thf(fact_439_succ__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% succ_correct
thf(fact_440_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y2 ) @ X2 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_441_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X2 ) @ X2 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_442_sumprop,axiom,
    vEBT_invar_vebt @ summary @ ( minus_minus_nat @ na @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sumprop
thf(fact_443_listlength,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ na @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% listlength
thf(fact_444_right__diff__distrib__numeral,axiom,
    ! [V: num,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( minus_4019991460397169231l_num1 @ B @ C ) )
      = ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ B ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_445_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_446_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_447_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_448_left__diff__distrib__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,V: num] :
      ( ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ A @ B ) @ ( numera7442385471795722001l_num1 @ V ) )
      = ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ A @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ B @ ( numera7442385471795722001l_num1 @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_449_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_450_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_451_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_452_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 ) )
        @ ^ [R: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_453_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 ) )
        @ ^ [R: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_454_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).

% both_member_options_ding
thf(fact_455_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X2 = Mi )
          | ( X2 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_456_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_457_power__add__numeral2,axiom,
    ! [A: uint32,M: num,N2: num,B: uint32] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_uint32 @ ( power_power_uint32 @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_uint32 @ ( power_power_uint32 @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_458_power__add__numeral2,axiom,
    ! [A: word_N3645301735248828278l_num1,M: num,N2: num,B: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_459_power__add__numeral2,axiom,
    ! [A: real,M: num,N2: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_460_power__add__numeral2,axiom,
    ! [A: rat,M: num,N2: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_461_power__add__numeral2,axiom,
    ! [A: nat,M: num,N2: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_462_power__add__numeral2,axiom,
    ! [A: int,M: num,N2: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_463_power__add__numeral,axiom,
    ! [A: uint32,M: num,N2: num] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_uint32 @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_uint32 @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_464_power__add__numeral,axiom,
    ! [A: word_N3645301735248828278l_num1,M: num,N2: num] :
      ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_465_power__add__numeral,axiom,
    ! [A: real,M: num,N2: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_466_power__add__numeral,axiom,
    ! [A: rat,M: num,N2: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_467_power__add__numeral,axiom,
    ! [A: nat,M: num,N2: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_468_power__add__numeral,axiom,
    ! [A: int,M: num,N2: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_469_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X2 )
      @ ^ [R: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_member @ T @ X2 ) ) ) ) ) ).

% vebt_memberi'_rf_abstr
thf(fact_470_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X2 ) ) ) ).

% vebt_inserti'_rf_abstr
thf(fact_471_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_472_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_473_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,G: heap_T2636463487746394924on_nat] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_7629718768684598413on_nat @ P @ G @ Q ) )
       => ( hoare_7629718768684598413on_nat @ P @ ( if_Hea5867803462524415986on_nat @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_474_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn,G: heap_Time_Heap_nat] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_3067605981109127869le_nat @ P @ G @ Q ) )
       => ( hoare_3067605981109127869le_nat @ P @ ( if_Hea2662716070787841314ap_nat @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_475_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn,G: heap_Time_Heap_o] :
      ( ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_hoare_triple_o @ P @ G @ Q ) )
       => ( hoare_hoare_triple_o @ P @ ( if_Heap_Time_Heap_o @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_476_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,G: heap_T8145700208782473153_VEBTi] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_1429296392585015714_VEBTi @ P @ G @ Q ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ ( if_Hea8453224502484754311_VEBTi @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_477_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T5738788834812785303t_unit,Q: product_unit > assn,G: heap_T5738788834812785303t_unit] :
      ( ( B
       => ( hoare_8945653483474564448t_unit @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_8945653483474564448t_unit @ P @ G @ Q ) )
       => ( hoare_8945653483474564448t_unit @ P @ ( if_Hea8138950348631371857t_unit @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_478_power2__commute,axiom,
    ! [X2: uint32,Y2: uint32] :
      ( ( power_power_uint32 @ ( minus_minus_uint32 @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_uint32 @ ( minus_minus_uint32 @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_479_power2__commute,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_2184487114949457152l_num1 @ ( minus_4019991460397169231l_num1 @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_480_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_481_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_482_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_483_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_484_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_485_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_486_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 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs )
             => ( ( ord_less_nat @ Z2 @ X )
               => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_487_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_488_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_489_power2__diff,axiom,
    ! [X2: uint32,Y2: uint32] :
      ( ( power_power_uint32 @ ( minus_minus_uint32 @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_uint32 @ ( plus_plus_uint32 @ ( power_power_uint32 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_uint32 @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_490_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_491_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_492_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_493_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_494_return__wp__rule,axiom,
    ! [Q: option_nat > assn,X2: option_nat] : ( hoare_7629718768684598413on_nat @ ( Q @ X2 ) @ ( heap_T3487192422709364219on_nat @ X2 ) @ Q ) ).

% return_wp_rule
thf(fact_495_return__wp__rule,axiom,
    ! [Q: nat > assn,X2: nat] : ( hoare_3067605981109127869le_nat @ ( Q @ X2 ) @ ( heap_Time_return_nat @ X2 ) @ Q ) ).

% return_wp_rule
thf(fact_496_return__wp__rule,axiom,
    ! [Q: $o > assn,X2: $o] : ( hoare_hoare_triple_o @ ( Q @ X2 ) @ ( heap_Time_return_o @ X2 ) @ Q ) ).

% return_wp_rule
thf(fact_497_return__wp__rule,axiom,
    ! [Q: vEBT_VEBTi > assn,X2: vEBT_VEBTi] : ( hoare_1429296392585015714_VEBTi @ ( Q @ X2 ) @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ Q ) ).

% return_wp_rule
thf(fact_498_return__wp__rule,axiom,
    ! [Q: product_unit > assn,X2: product_unit] : ( hoare_8945653483474564448t_unit @ ( Q @ X2 ) @ ( heap_T7507251653302230130t_unit @ X2 ) @ Q ) ).

% return_wp_rule
thf(fact_499_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R2: nat > assn,G: nat > heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ F @ R2 )
     => ( ! [X3: nat] : ( hoare_3067605981109127869le_nat @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_3067605981109127869le_nat @ P @ ( heap_T7049098217575491753at_nat @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_500_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R2: nat > assn,G: nat > heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ F @ R2 )
     => ( ! [X3: nat] : ( hoare_hoare_triple_o @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_hoare_triple_o @ P @ ( heap_Time_bind_nat_o @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_501_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R2: nat > assn,G: nat > heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ F @ R2 )
     => ( ! [X3: nat] : ( hoare_1429296392585015714_VEBTi @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_1429296392585015714_VEBTi @ P @ ( heap_T844888390831797134_VEBTi @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_502_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,R2: nat > assn,G: nat > heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ F @ R2 )
     => ( ! [X3: nat] : ( hoare_8945653483474564448t_unit @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_8945653483474564448t_unit @ P @ ( heap_T2412598413086283380t_unit @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_503_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R2: $o > assn,G: $o > heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_hoare_triple_o @ P @ F @ R2 )
     => ( ! [X3: $o] : ( hoare_3067605981109127869le_nat @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_3067605981109127869le_nat @ P @ ( heap_Time_bind_o_nat @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_504_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R2: $o > assn,G: $o > heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ P @ F @ R2 )
     => ( ! [X3: $o] : ( hoare_hoare_triple_o @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_hoare_triple_o @ P @ ( heap_Time_bind_o_o @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_505_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R2: $o > assn,G: $o > heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_hoare_triple_o @ P @ F @ R2 )
     => ( ! [X3: $o] : ( hoare_1429296392585015714_VEBTi @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_1429296392585015714_VEBTi @ P @ ( heap_T5998771940306268294_VEBTi @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_506_bind__rule,axiom,
    ! [P: assn,F: heap_Time_Heap_o,R2: $o > assn,G: $o > heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ( hoare_hoare_triple_o @ P @ F @ R2 )
     => ( ! [X3: $o] : ( hoare_8945653483474564448t_unit @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_8945653483474564448t_unit @ P @ ( heap_T2991970491176150908t_unit @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_507_bind__rule,axiom,
    ! [P: assn,F: heap_T8145700208782473153_VEBTi,R2: vEBT_VEBTi > assn,G: vEBT_VEBTi > heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ F @ R2 )
     => ( ! [X3: vEBT_VEBTi] : ( hoare_3067605981109127869le_nat @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_3067605981109127869le_nat @ P @ ( heap_T5387808048883414158Ti_nat @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_508_bind__rule,axiom,
    ! [P: assn,F: heap_T8145700208782473153_VEBTi,R2: vEBT_VEBTi > assn,G: vEBT_VEBTi > heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ F @ R2 )
     => ( ! [X3: vEBT_VEBTi] : ( hoare_hoare_triple_o @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_hoare_triple_o @ P @ ( heap_T3040810144269856602EBTi_o @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_509_VEBTi_Oexhaust,axiom,
    ! [Y2: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] :
          ( Y2
         != ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X222: $o] :
            ( Y2
           != ( vEBT_Leafi @ X212 @ X222 ) ) ) ).

% VEBTi.exhaust
thf(fact_510_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X22: $o] :
      ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leafi @ X21 @ X22 ) ) ).

% VEBTi.distinct(1)
thf(fact_511_frame__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,R2: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ R2 ) @ C
        @ ^ [X: option_nat] : ( times_times_assn @ ( Q @ X ) @ R2 ) ) ) ).

% frame_rule
thf(fact_512_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,R2: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ R2 ) @ C
        @ ^ [X: nat] : ( times_times_assn @ ( Q @ X ) @ R2 ) ) ) ).

% frame_rule
thf(fact_513_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,R2: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ R2 ) @ C
        @ ^ [X: $o] : ( times_times_assn @ ( Q @ X ) @ R2 ) ) ) ).

% frame_rule
thf(fact_514_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R2: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ R2 ) @ C
        @ ^ [X: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X ) @ R2 ) ) ) ).

% frame_rule
thf(fact_515_frame__rule,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn,R2: assn] :
      ( ( hoare_8945653483474564448t_unit @ P @ C @ Q )
     => ( hoare_8945653483474564448t_unit @ ( times_times_assn @ P @ R2 ) @ C
        @ ^ [X: product_unit] : ( times_times_assn @ ( Q @ X ) @ R2 ) ) ) ).

% frame_rule
thf(fact_516_norm__pre__ex__rule,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ! [X3: list_VEBT_VEBTi] : ( hoare_7629718768684598413on_nat @ ( P @ X3 ) @ F @ Q )
     => ( hoare_7629718768684598413on_nat @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q ) ) ).

% norm_pre_ex_rule
thf(fact_517_norm__pre__ex__rule,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ! [X3: list_VEBT_VEBTi] : ( hoare_3067605981109127869le_nat @ ( P @ X3 ) @ F @ Q )
     => ( hoare_3067605981109127869le_nat @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q ) ) ).

% norm_pre_ex_rule
thf(fact_518_norm__pre__ex__rule,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ! [X3: list_VEBT_VEBTi] : ( hoare_hoare_triple_o @ ( P @ X3 ) @ F @ Q )
     => ( hoare_hoare_triple_o @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q ) ) ).

% norm_pre_ex_rule
thf(fact_519_norm__pre__ex__rule,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ! [X3: list_VEBT_VEBTi] : ( hoare_1429296392585015714_VEBTi @ ( P @ X3 ) @ F @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q ) ) ).

% norm_pre_ex_rule
thf(fact_520_norm__pre__ex__rule,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ! [X3: list_VEBT_VEBTi] : ( hoare_8945653483474564448t_unit @ ( P @ X3 ) @ F @ Q )
     => ( hoare_8945653483474564448t_unit @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q ) ) ).

% norm_pre_ex_rule
thf(fact_521_post__exI__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > list_VEBT_VEBTi > assn,X2: list_VEBT_VEBTi] :
      ( ( hoare_7629718768684598413on_nat @ P @ C
        @ ^ [R: option_nat] : ( Q @ R @ X2 ) )
     => ( hoare_7629718768684598413on_nat @ P @ C
        @ ^ [R: option_nat] : ( ex_ass463751140784270563_VEBTi @ ( Q @ R ) ) ) ) ).

% post_exI_rule
thf(fact_522_post__exI__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > list_VEBT_VEBTi > assn,X2: list_VEBT_VEBTi] :
      ( ( hoare_3067605981109127869le_nat @ P @ C
        @ ^ [R: nat] : ( Q @ R @ X2 ) )
     => ( hoare_3067605981109127869le_nat @ P @ C
        @ ^ [R: nat] : ( ex_ass463751140784270563_VEBTi @ ( Q @ R ) ) ) ) ).

% post_exI_rule
thf(fact_523_post__exI__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > list_VEBT_VEBTi > assn,X2: list_VEBT_VEBTi] :
      ( ( hoare_hoare_triple_o @ P @ C
        @ ^ [R: $o] : ( Q @ R @ X2 ) )
     => ( hoare_hoare_triple_o @ P @ C
        @ ^ [R: $o] : ( ex_ass463751140784270563_VEBTi @ ( Q @ R ) ) ) ) ).

% post_exI_rule
thf(fact_524_post__exI__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > list_VEBT_VEBTi > assn,X2: list_VEBT_VEBTi] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C
        @ ^ [R: vEBT_VEBTi] : ( Q @ R @ X2 ) )
     => ( hoare_1429296392585015714_VEBTi @ P @ C
        @ ^ [R: vEBT_VEBTi] : ( ex_ass463751140784270563_VEBTi @ ( Q @ R ) ) ) ) ).

% post_exI_rule
thf(fact_525_post__exI__rule,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > list_VEBT_VEBTi > assn,X2: list_VEBT_VEBTi] :
      ( ( hoare_8945653483474564448t_unit @ P @ C
        @ ^ [R: product_unit] : ( Q @ R @ X2 ) )
     => ( hoare_8945653483474564448t_unit @ P @ C
        @ ^ [R: product_unit] : ( ex_ass463751140784270563_VEBTi @ ( Q @ R ) ) ) ) ).

% post_exI_rule
thf(fact_526_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_527_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_528_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_529_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_530_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ( B
       => ( hoare_8945653483474564448t_unit @ P @ F @ Q ) )
     => ( hoare_8945653483474564448t_unit @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_531_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_532_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_533_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_534_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_535_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ( B
       => ( hoare_8945653483474564448t_unit @ one_one_assn @ F @ Q ) )
     => ( hoare_8945653483474564448t_unit @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_536_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_Time_Heap_nat,Q: nat > assn,Fs: nat > heap_Time_Heap_nat] :
      ( ( ( V = none_nat )
       => ( hoare_3067605981109127869le_nat @ P @ Fn @ Q ) )
     => ( ! [X3: nat] :
            ( ( V
              = ( some_nat @ X3 ) )
           => ( hoare_3067605981109127869le_nat @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_3067605981109127869le_nat @ P @ ( case_o6609685678014844897at_nat @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_537_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_Time_Heap_nat,Q: nat > assn,Fs: num > heap_Time_Heap_nat] :
      ( ( ( V = none_num )
       => ( hoare_3067605981109127869le_nat @ P @ Fn @ Q ) )
     => ( ! [X3: num] :
            ( ( V
              = ( some_num @ X3 ) )
           => ( hoare_3067605981109127869le_nat @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_3067605981109127869le_nat @ P @ ( case_o3167017464170623531at_num @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_538_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_Time_Heap_o,Q: $o > assn,Fs: nat > heap_Time_Heap_o] :
      ( ( ( V = none_nat )
       => ( hoare_hoare_triple_o @ P @ Fn @ Q ) )
     => ( ! [X3: nat] :
            ( ( V
              = ( some_nat @ X3 ) )
           => ( hoare_hoare_triple_o @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_hoare_triple_o @ P @ ( case_o6892868863119666303_o_nat @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_539_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_Time_Heap_o,Q: $o > assn,Fs: num > heap_Time_Heap_o] :
      ( ( ( V = none_num )
       => ( hoare_hoare_triple_o @ P @ Fn @ Q ) )
     => ( ! [X3: num] :
            ( ( V
              = ( some_num @ X3 ) )
           => ( hoare_hoare_triple_o @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_hoare_triple_o @ P @ ( case_o3450200649275444937_o_num @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_540_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Fs: nat > heap_T8145700208782473153_VEBTi] :
      ( ( ( V = none_nat )
       => ( hoare_1429296392585015714_VEBTi @ P @ Fn @ Q ) )
     => ( ! [X3: nat] :
            ( ( V
              = ( some_nat @ X3 ) )
           => ( hoare_1429296392585015714_VEBTi @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ ( case_o3780387683879180358Ti_nat @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_541_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Fs: num > heap_T8145700208782473153_VEBTi] :
      ( ( ( V = none_num )
       => ( hoare_1429296392585015714_VEBTi @ P @ Fn @ Q ) )
     => ( ! [X3: num] :
            ( ( V
              = ( some_num @ X3 ) )
           => ( hoare_1429296392585015714_VEBTi @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ ( case_o337719470034958992Ti_num @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_542_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_T5738788834812785303t_unit,Q: product_unit > assn,Fs: nat > heap_T5738788834812785303t_unit] :
      ( ( ( V = none_nat )
       => ( hoare_8945653483474564448t_unit @ P @ Fn @ Q ) )
     => ( ! [X3: nat] :
            ( ( V
              = ( some_nat @ X3 ) )
           => ( hoare_8945653483474564448t_unit @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_8945653483474564448t_unit @ P @ ( case_o8044209130310455284it_nat @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_543_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_T5738788834812785303t_unit,Q: product_unit > assn,Fs: num > heap_T5738788834812785303t_unit] :
      ( ( ( V = none_num )
       => ( hoare_8945653483474564448t_unit @ P @ Fn @ Q ) )
     => ( ! [X3: num] :
            ( ( V
              = ( some_num @ X3 ) )
           => ( hoare_8945653483474564448t_unit @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_8945653483474564448t_unit @ P @ ( case_o4601540916466233918it_num @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_544_case__option__rule,axiom,
    ! [V: option_nat,P: assn,Fn: heap_T2636463487746394924on_nat,Q: option_nat > assn,Fs: nat > heap_T2636463487746394924on_nat] :
      ( ( ( V = none_nat )
       => ( hoare_7629718768684598413on_nat @ P @ Fn @ Q ) )
     => ( ! [X3: nat] :
            ( ( V
              = ( some_nat @ X3 ) )
           => ( hoare_7629718768684598413on_nat @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_7629718768684598413on_nat @ P @ ( case_o2256915875499652529at_nat @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_545_case__option__rule,axiom,
    ! [V: option_num,P: assn,Fn: heap_T2636463487746394924on_nat,Q: option_nat > assn,Fs: num > heap_T2636463487746394924on_nat] :
      ( ( ( V = none_num )
       => ( hoare_7629718768684598413on_nat @ P @ Fn @ Q ) )
     => ( ! [X3: num] :
            ( ( V
              = ( some_num @ X3 ) )
           => ( hoare_7629718768684598413on_nat @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_7629718768684598413on_nat @ P @ ( case_o8037619698510206971at_num @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_546_return__sp__rule,axiom,
    ! [P: assn,X2: option_nat] :
      ( hoare_7629718768684598413on_nat @ P @ ( heap_T3487192422709364219on_nat @ X2 )
      @ ^ [R: option_nat] : ( times_times_assn @ P @ ( pure_assn @ ( R = X2 ) ) ) ) ).

% return_sp_rule
thf(fact_547_return__sp__rule,axiom,
    ! [P: assn,X2: nat] :
      ( hoare_3067605981109127869le_nat @ P @ ( heap_Time_return_nat @ X2 )
      @ ^ [R: nat] : ( times_times_assn @ P @ ( pure_assn @ ( R = X2 ) ) ) ) ).

% return_sp_rule
thf(fact_548_return__sp__rule,axiom,
    ! [P: assn,X2: $o] :
      ( hoare_hoare_triple_o @ P @ ( heap_Time_return_o @ X2 )
      @ ^ [R: $o] : ( times_times_assn @ P @ ( pure_assn @ ( R = X2 ) ) ) ) ).

% return_sp_rule
thf(fact_549_return__sp__rule,axiom,
    ! [P: assn,X2: vEBT_VEBTi] :
      ( hoare_1429296392585015714_VEBTi @ P @ ( heap_T3630416162098727440_VEBTi @ X2 )
      @ ^ [R: vEBT_VEBTi] : ( times_times_assn @ P @ ( pure_assn @ ( R = X2 ) ) ) ) ).

% return_sp_rule
thf(fact_550_return__sp__rule,axiom,
    ! [P: assn,X2: product_unit] :
      ( hoare_8945653483474564448t_unit @ P @ ( heap_T7507251653302230130t_unit @ X2 )
      @ ^ [R: product_unit] : ( times_times_assn @ P @ ( pure_assn @ ( R = X2 ) ) ) ) ).

% return_sp_rule
thf(fact_551_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_552_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi ) ) ) ).

% vebt_minti.simps(3)
thf(fact_553_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma ) ) ) ).

% vebt_maxti.simps(3)
thf(fact_554_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_555_vebt__succ_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_556_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N: nat,TreeList3: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).

% in_children_def
thf(fact_557_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_558_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_559_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_560_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_561_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_562_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_563_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_564_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_565_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_566_option_Ocollapse,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( ( some_P2407035485129114418it_nat @ ( the_Pr3501439614016493281it_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_567_option_Ocollapse,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( ( some_P468703482102919278it_nat @ ( the_Pr5838048819577852031it_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_568_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_569_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_570_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_571_highsimp,axiom,
    ! [X2: nat,N2: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_high @ X2 @ N2 ) )
      = ( vEBT_VEBT_highi @ X2 @ N2 ) ) ).

% highsimp
thf(fact_572_lowsimp,axiom,
    ! [X2: nat,N2: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_low @ X2 @ N2 ) )
      = ( vEBT_VEBT_lowi @ X2 @ N2 ) ) ).

% lowsimp
thf(fact_573_semiring__norm_I87_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(87)
thf(fact_574_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( ( suc @ X23 )
        = ( suc @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% nat.inject
thf(fact_575_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_576_option_Oinject,axiom,
    ! [X23: product_prod_nat_nat,Y23: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X23 )
        = ( some_P7363390416028606310at_nat @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% option.inject
thf(fact_577_option_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( ( some_nat @ X23 )
        = ( some_nat @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% option.inject
thf(fact_578_option_Oinject,axiom,
    ! [X23: num,Y23: num] :
      ( ( ( some_num @ X23 )
        = ( some_num @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% option.inject
thf(fact_579_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_580_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_581_semiring__norm_I83_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit0 @ N2 ) ) ).

% semiring_norm(83)
thf(fact_582_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_583_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_584_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_585_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_586_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_587_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_588_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_589_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_590_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_591_not__Some__eq2,axiom,
    ! [V: option8051342751916580710nteger] :
      ( ( ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
            ( V
           != ( some_P1462369734362851057nteger @ ( produc8603105652947943368nteger @ X @ Y ) ) ) )
      = ( V = none_P4442379456014020469nteger ) ) ).

% not_Some_eq2
thf(fact_592_not__Some__eq2,axiom,
    ! [V: option5190343406534369742et_nat] :
      ( ( ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
            ( V
           != ( some_P750831030444334937et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) )
      = ( V = none_P4972525538344268765et_nat ) ) ).

% not_Some_eq2
thf(fact_593_not__Some__eq2,axiom,
    ! [V: option2860828798490689354et_nat] :
      ( ( ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
            ( V
           != ( some_P1630309045189364437et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) )
      = ( V = none_P199884684680593241et_nat ) ) ).

% not_Some_eq2
thf(fact_594_not__Some__eq2,axiom,
    ! [V: option7541221861074943443nt_int] :
      ( ( ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
            ( V
           != ( some_P2355398578364412894nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) )
      = ( V = none_P1286213070022356066nt_int ) ) ).

% not_Some_eq2
thf(fact_595_not__Some__eq2,axiom,
    ! [V: option4256020574406277085nt_int] :
      ( ( ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
            ( V
           != ( some_P7455497367792166888nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) )
      = ( V = none_P3773570700014501484nt_int ) ) ).

% not_Some_eq2
thf(fact_596_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_597_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_598_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_599_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_600_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_601_not__None__eq,axiom,
    ! [X2: option2621746655072343315it_nat] :
      ( ( X2 != none_P1551326421579882414it_nat )
      = ( ? [Y: produc120671012495760973it_nat] :
            ( X2
            = ( some_P2407035485129114418it_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_602_not__None__eq,axiom,
    ! [X2: option7339022715339332451it_nat] :
      ( ( X2 != none_P7668321371905463026it_nat )
      = ( ? [Y: produc8047831477865546771it_nat] :
            ( X2
            = ( some_P468703482102919278it_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_603_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_604_not__None__eq,axiom,
    ! [X2: option_nat] :
      ( ( X2 != none_nat )
      = ( ? [Y: nat] :
            ( X2
            = ( some_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_605_not__None__eq,axiom,
    ! [X2: option_num] :
      ( ( X2 != none_num )
      = ( ? [Y: num] :
            ( X2
            = ( some_num @ Y ) ) ) ) ).

% not_None_eq
thf(fact_606_not__Some__eq,axiom,
    ! [X2: option2621746655072343315it_nat] :
      ( ( ! [Y: produc120671012495760973it_nat] :
            ( X2
           != ( some_P2407035485129114418it_nat @ Y ) ) )
      = ( X2 = none_P1551326421579882414it_nat ) ) ).

% not_Some_eq
thf(fact_607_not__Some__eq,axiom,
    ! [X2: option7339022715339332451it_nat] :
      ( ( ! [Y: produc8047831477865546771it_nat] :
            ( X2
           != ( some_P468703482102919278it_nat @ Y ) ) )
      = ( X2 = none_P7668321371905463026it_nat ) ) ).

% not_Some_eq
thf(fact_608_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_609_not__Some__eq,axiom,
    ! [X2: option_nat] :
      ( ( ! [Y: nat] :
            ( X2
           != ( some_nat @ Y ) ) )
      = ( X2 = none_nat ) ) ).

% not_Some_eq
thf(fact_610_not__Some__eq,axiom,
    ! [X2: option_num] :
      ( ( ! [Y: num] :
            ( X2
           != ( some_num @ Y ) ) )
      = ( X2 = none_num ) ) ).

% not_Some_eq
thf(fact_611_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_612_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_613_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_614_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_615_semiring__norm_I12_J,axiom,
    ! [N2: num] :
      ( ( times_times_num @ one @ N2 )
      = N2 ) ).

% semiring_norm(12)
thf(fact_616_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_617_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_618_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_619_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).

% semiring_norm(68)
thf(fact_620_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_621_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_622_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_623_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_624_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_625_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_626_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_627_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_628_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 ) ).

% vebt_member.simps(2)
thf(fact_629_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_set_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_630_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_rat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_631_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_num @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_632_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_nat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_633_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_634_pairself_Ocases,axiom,
    ! [X2: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,A3: int,B2: int] :
          ( X2
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ A3 @ B2 ) ) ) ).

% pairself.cases
thf(fact_635_bex2I,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,S2: set_Pr1281608226676607948nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S2 )
     => ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_636_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S2 )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_637_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S2 )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_638_bex2I,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr9222295170931077689nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S2 )
     => ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_639_bex2I,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr1872883991513573699nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S2 )
     => ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_640_Suc__inject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
     => ( X2 = Y2 ) ) ).

% Suc_inject
thf(fact_641_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_642_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_643_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_644_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_645_less__not__refl3,axiom,
    ! [S3: nat,T: nat] :
      ( ( ord_less_nat @ S3 @ T )
     => ( S3 != T ) ) ).

% less_not_refl3
thf(fact_646_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_647_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_648_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_649_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_650_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_651_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_652_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_653_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_654_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_655_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_656_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_657_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_658_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_659_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_660_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_661_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_662_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_663_VEBT__internal_OminNulli_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ( X2
       != ( vEBT_Leafi @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leafi @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leafi @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                ( X2
               != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                  ( X2
                 != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ).

% VEBT_internal.minNulli.cases
thf(fact_664_vebt__minti_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ! [A3: $o,B2: $o] :
          ( X2
         != ( vEBT_Leafi @ A3 @ B2 ) )
     => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
            ( X2
           != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
              ( X2
             != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_minti.cases
thf(fact_665_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_maxti.simps(2)
thf(fact_666_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_minti.simps(2)
thf(fact_667_option_Ocase__distrib,axiom,
    ! [H2: num > num,F1: num,F22: num > num,Option: option_num] :
      ( ( H2 @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_668_option_Ocase__distrib,axiom,
    ! [H2: num > int,F1: num,F22: num > num,Option: option_num] :
      ( ( H2 @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_669_option_Ocase__distrib,axiom,
    ! [H2: int > num,F1: int,F22: num > int,Option: option_num] :
      ( ( H2 @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_670_option_Ocase__distrib,axiom,
    ! [H2: int > int,F1: int,F22: num > int,Option: option_num] :
      ( ( H2 @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_671_option_Ocase__distrib,axiom,
    ! [H2: option_num > num,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H2 @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_672_option_Ocase__distrib,axiom,
    ! [H2: option_num > int,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H2 @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_673_option_Ocase__distrib,axiom,
    ! [H2: num > option_num,F1: num,F22: num > num,Option: option_num] :
      ( ( H2 @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_674_option_Ocase__distrib,axiom,
    ! [H2: int > option_num,F1: int,F22: num > int,Option: option_num] :
      ( ( H2 @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_675_option_Ocase__distrib,axiom,
    ! [H2: option_num > option_num,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H2 @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_676_option_Ocase__distrib,axiom,
    ! [H2: $o > $o,F1: $o,F22: product_prod_nat_nat > $o,Option: option4927543243414619207at_nat] :
      ( ( H2 @ ( case_o184042715313410164at_nat @ F1 @ F22 @ Option ) )
      = ( case_o184042715313410164at_nat @ ( H2 @ F1 )
        @ ^ [X: product_prod_nat_nat] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_677_z1pdiv2,axiom,
    ! [B: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = B ) ).

% z1pdiv2
thf(fact_678_le__some__optE,axiom,
    ! [M: set_int,X2: option_set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ M ) @ X2 )
     => ~ ! [M4: set_int] :
            ( ( X2
              = ( some_set_int @ M4 ) )
           => ~ ( ord_less_eq_set_int @ M @ M4 ) ) ) ).

% le_some_optE
thf(fact_679_le__some__optE,axiom,
    ! [M: rat,X2: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M ) @ X2 )
     => ~ ! [M4: rat] :
            ( ( X2
              = ( some_rat @ M4 ) )
           => ~ ( ord_less_eq_rat @ M @ M4 ) ) ) ).

% le_some_optE
thf(fact_680_le__some__optE,axiom,
    ! [M: num,X2: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X2 )
     => ~ ! [M4: num] :
            ( ( X2
              = ( some_num @ M4 ) )
           => ~ ( ord_less_eq_num @ M @ M4 ) ) ) ).

% le_some_optE
thf(fact_681_le__some__optE,axiom,
    ! [M: nat,X2: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X2 )
     => ~ ! [M4: nat] :
            ( ( X2
              = ( some_nat @ M4 ) )
           => ~ ( ord_less_eq_nat @ M @ M4 ) ) ) ).

% le_some_optE
thf(fact_682_le__some__optE,axiom,
    ! [M: int,X2: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X2 )
     => ~ ! [M4: int] :
            ( ( X2
              = ( some_int @ M4 ) )
           => ~ ( ord_less_eq_int @ M @ M4 ) ) ) ).

% le_some_optE
thf(fact_683_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_684_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_685_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_686_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_687_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_688_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_689_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N2 )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_690_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_691_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_692_Nat_OAll__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N2 )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
           => ( P @ I2 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_693_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_694_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_695_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_696_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_697_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I3 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_698_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_699_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_700_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_701_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_702_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_703_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_704_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_705_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_706_Suc__le__D,axiom,
    ! [N2: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
     => ? [M2: nat] :
          ( M6
          = ( suc @ M2 ) ) ) ).

% Suc_le_D
thf(fact_707_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_708_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_709_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_710_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_711_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_712_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
           => ( R2 @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_713_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_714_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_715_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_716_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_717_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_718_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_719_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_720_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_721_exists__leI,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [N6: nat] :
            ( ( ord_less_nat @ N6 @ N2 )
           => ~ ( P @ N6 ) )
       => ( P @ N2 ) )
     => ? [N7: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
          & ( P @ N7 ) ) ) ).

% exists_leI
thf(fact_722_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M7: nat,N: nat] :
          ( ( ord_less_eq_nat @ M7 @ N )
          & ( M7 != N ) ) ) ) ).

% nat_less_le
thf(fact_723_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_724_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M7: nat,N: nat] :
          ( ( ord_less_nat @ M7 @ N )
          | ( M7 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_725_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_726_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_727_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_728_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_729_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_730_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_731_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_732_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_733_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_734_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_735_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_736_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_737_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N4: nat] :
          ( L
          = ( plus_plus_nat @ K @ N4 ) ) ) ).

% le_Suc_ex
thf(fact_738_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_739_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_740_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_741_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_742_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M7: nat,N: nat] :
        ? [K4: nat] :
          ( N
          = ( plus_plus_nat @ M7 @ K4 ) ) ) ) ).

% nat_le_iff_add
thf(fact_743_fstE,axiom,
    ! [X2: produc1908205239877642774nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o] :
      ( ( X2
        = ( produc8603105652947943368nteger @ A @ B ) )
     => ( ( P @ ( produc7822682618958472924nteger @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_744_fstE,axiom,
    ! [X2: produc3925858234332021118et_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( X2
        = ( produc5001842942810119800et_nat @ A @ B ) )
     => ( ( P @ ( produc995936583742144908et_nat @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_745_fstE,axiom,
    ! [X2: produc2732055786443039994et_nat,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( X2
        = ( produc2245416461498447860et_nat @ A @ B ) )
     => ( ( P @ ( produc180342877477747464et_nat @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_746_fstE,axiom,
    ! [X2: produc2285326912895808259nt_int,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o] :
      ( ( X2
        = ( produc5700946648718959541nt_int @ A @ B ) )
     => ( ( P @ ( produc6816164490631068361nt_int @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_747_fstE,axiom,
    ! [X2: produc7773217078559923341nt_int,A: int > option6357759511663192854e_term,B: product_prod_int_int,P: ( int > option6357759511663192854e_term ) > $o] :
      ( ( X2
        = ( produc4305682042979456191nt_int @ A @ B ) )
     => ( ( P @ ( produc6230002227079971283nt_int @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_748_fstE,axiom,
    ! [X2: produc3658429121746597890et_nat,A: heap_e7401611519738050253t_unit,B: set_nat,P: heap_e7401611519738050253t_unit > $o] :
      ( ( X2
        = ( produc7507926704131184380et_nat @ A @ B ) )
     => ( ( P @ ( produc1824681642469235216et_nat @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_749_fstE,axiom,
    ! [X2: produc6575502325842934193n_assn,A: assn,B: assn,P: assn > $o] :
      ( ( X2
        = ( produc118845697133431529n_assn @ A @ B ) )
     => ( ( P @ ( produc9167289414957590229n_assn @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_750_fstE,axiom,
    ! [X2: produc827990862158126777uint32,A: uint32,B: uint32,P: uint32 > $o] :
      ( ( X2
        = ( produc1400373151660368625uint32 @ A @ B ) )
     => ( ( P @ ( produc9004433772639906525uint32 @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_751_fstE,axiom,
    ! [X2: product_prod_nat_nat,A: nat,B: nat,P: nat > $o] :
      ( ( X2
        = ( product_Pair_nat_nat @ A @ B ) )
     => ( ( P @ ( product_fst_nat_nat @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_752_fstE,axiom,
    ! [X2: product_prod_int_int,A: int,B: int,P: int > $o] :
      ( ( X2
        = ( product_Pair_int_int @ A @ B ) )
     => ( ( P @ ( product_fst_int_int @ X2 ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_753_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_754_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_755_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_756_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_757_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_758_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_759_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_760_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_761_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_762_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_763_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_764_option_Odistinct_I1_J,axiom,
    ! [X23: produc120671012495760973it_nat] :
      ( none_P1551326421579882414it_nat
     != ( some_P2407035485129114418it_nat @ X23 ) ) ).

% option.distinct(1)
thf(fact_765_option_Odistinct_I1_J,axiom,
    ! [X23: produc8047831477865546771it_nat] :
      ( none_P7668321371905463026it_nat
     != ( some_P468703482102919278it_nat @ X23 ) ) ).

% option.distinct(1)
thf(fact_766_option_Odistinct_I1_J,axiom,
    ! [X23: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X23 ) ) ).

% option.distinct(1)
thf(fact_767_option_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( none_nat
     != ( some_nat @ X23 ) ) ).

% option.distinct(1)
thf(fact_768_option_Odistinct_I1_J,axiom,
    ! [X23: num] :
      ( none_num
     != ( some_num @ X23 ) ) ).

% option.distinct(1)
thf(fact_769_option_OdiscI,axiom,
    ! [Option: option2621746655072343315it_nat,X23: produc120671012495760973it_nat] :
      ( ( Option
        = ( some_P2407035485129114418it_nat @ X23 ) )
     => ( Option != none_P1551326421579882414it_nat ) ) ).

% option.discI
thf(fact_770_option_OdiscI,axiom,
    ! [Option: option7339022715339332451it_nat,X23: produc8047831477865546771it_nat] :
      ( ( Option
        = ( some_P468703482102919278it_nat @ X23 ) )
     => ( Option != none_P7668321371905463026it_nat ) ) ).

% option.discI
thf(fact_771_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X23: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X23 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_772_option_OdiscI,axiom,
    ! [Option: option_nat,X23: nat] :
      ( ( Option
        = ( some_nat @ X23 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_773_option_OdiscI,axiom,
    ! [Option: option_num,X23: num] :
      ( ( Option
        = ( some_num @ X23 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_774_option_Oexhaust,axiom,
    ! [Y2: option2621746655072343315it_nat] :
      ( ( Y2 != none_P1551326421579882414it_nat )
     => ~ ! [X24: produc120671012495760973it_nat] :
            ( Y2
           != ( some_P2407035485129114418it_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_775_option_Oexhaust,axiom,
    ! [Y2: option7339022715339332451it_nat] :
      ( ( Y2 != none_P7668321371905463026it_nat )
     => ~ ! [X24: produc8047831477865546771it_nat] :
            ( Y2
           != ( some_P468703482102919278it_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_776_option_Oexhaust,axiom,
    ! [Y2: option4927543243414619207at_nat] :
      ( ( Y2 != none_P5556105721700978146at_nat )
     => ~ ! [X24: product_prod_nat_nat] :
            ( Y2
           != ( some_P7363390416028606310at_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_777_option_Oexhaust,axiom,
    ! [Y2: option_nat] :
      ( ( Y2 != none_nat )
     => ~ ! [X24: nat] :
            ( Y2
           != ( some_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_778_option_Oexhaust,axiom,
    ! [Y2: option_num] :
      ( ( Y2 != none_num )
     => ~ ! [X24: num] :
            ( Y2
           != ( some_num @ X24 ) ) ) ).

% option.exhaust
thf(fact_779_split__option__ex,axiom,
    ( ( ^ [P2: option2621746655072343315it_nat > $o] :
        ? [X4: option2621746655072343315it_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option2621746655072343315it_nat > $o] :
          ( ( P3 @ none_P1551326421579882414it_nat )
          | ? [X: produc120671012495760973it_nat] : ( P3 @ ( some_P2407035485129114418it_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_780_split__option__ex,axiom,
    ( ( ^ [P2: option7339022715339332451it_nat > $o] :
        ? [X4: option7339022715339332451it_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option7339022715339332451it_nat > $o] :
          ( ( P3 @ none_P7668321371905463026it_nat )
          | ? [X: produc8047831477865546771it_nat] : ( P3 @ ( some_P468703482102919278it_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_781_split__option__ex,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_782_split__option__ex,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ? [X4: option_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_783_split__option__ex,axiom,
    ( ( ^ [P2: option_num > $o] :
        ? [X4: option_num] : ( P2 @ X4 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_784_split__option__all,axiom,
    ( ( ^ [P2: option2621746655072343315it_nat > $o] :
        ! [X4: option2621746655072343315it_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option2621746655072343315it_nat > $o] :
          ( ( P3 @ none_P1551326421579882414it_nat )
          & ! [X: produc120671012495760973it_nat] : ( P3 @ ( some_P2407035485129114418it_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_785_split__option__all,axiom,
    ( ( ^ [P2: option7339022715339332451it_nat > $o] :
        ! [X4: option7339022715339332451it_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option7339022715339332451it_nat > $o] :
          ( ( P3 @ none_P7668321371905463026it_nat )
          & ! [X: produc8047831477865546771it_nat] : ( P3 @ ( some_P468703482102919278it_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_786_split__option__all,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_787_split__option__all,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ! [X4: option_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_788_split__option__all,axiom,
    ( ( ^ [P2: option_num > $o] :
        ! [X4: option_num] : ( P2 @ X4 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_789_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: nat,B2: nat] :
              ( ( X2
                = ( some_nat @ A3 ) )
             => ( ( Y2
                  = ( some_nat @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_790_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: nat,B2: num] :
              ( ( X2
                = ( some_nat @ A3 ) )
             => ( ( Y2
                  = ( some_num @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_791_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: num,B2: nat] :
              ( ( X2
                = ( some_num @ A3 ) )
             => ( ( Y2
                  = ( some_nat @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_792_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: num,B2: num] :
              ( ( X2
                = ( some_num @ A3 ) )
             => ( ( Y2
                  = ( some_num @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_793_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: product_prod_nat_nat,B2: nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y2
                  = ( some_nat @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_794_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: product_prod_nat_nat,B2: num] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y2
                  = ( some_num @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_795_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: nat,B2: product_prod_nat_nat] :
              ( ( X2
                = ( some_nat @ A3 ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_796_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: num,B2: product_prod_nat_nat] :
              ( ( X2
                = ( some_num @ A3 ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_797_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_798_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 ) )
       => ( ! [A3: produc8047831477865546771it_nat,B2: nat] :
              ( ( X2
                = ( some_P468703482102919278it_nat @ A3 ) )
             => ( ( Y2
                  = ( some_nat @ B2 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_799_sndE,axiom,
    ! [X2: produc1908205239877642774nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,P: produc8923325533196201883nteger > $o] :
      ( ( X2
        = ( produc8603105652947943368nteger @ A @ B ) )
     => ( ( P @ ( produc7856867400915047194nteger @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_800_sndE,axiom,
    ! [X2: produc3925858234332021118et_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: produc3658429121746597890et_nat > $o] :
      ( ( X2
        = ( produc5001842942810119800et_nat @ A @ B ) )
     => ( ( P @ ( produc4011572625026189258et_nat @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_801_sndE,axiom,
    ! [X2: produc2732055786443039994et_nat,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: produc3925858234332021118et_nat > $o] :
      ( ( X2
        = ( produc2245416461498447860et_nat @ A @ B ) )
     => ( ( P @ ( produc5374455773327741254et_nat @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_802_sndE,axiom,
    ! [X2: produc2285326912895808259nt_int,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( X2
        = ( produc5700946648718959541nt_int @ A @ B ) )
     => ( ( P @ ( produc7328097813583171335nt_int @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_803_sndE,axiom,
    ! [X2: produc7773217078559923341nt_int,A: int > option6357759511663192854e_term,B: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( X2
        = ( produc4305682042979456191nt_int @ A @ B ) )
     => ( ( P @ ( produc3162348030201620241nt_int @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_804_sndE,axiom,
    ! [X2: product_prod_nat_nat,A: nat,B: nat,P: nat > $o] :
      ( ( X2
        = ( product_Pair_nat_nat @ A @ B ) )
     => ( ( P @ ( product_snd_nat_nat @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_805_sndE,axiom,
    ! [X2: produc3658429121746597890et_nat,A: heap_e7401611519738050253t_unit,B: set_nat,P: set_nat > $o] :
      ( ( X2
        = ( produc7507926704131184380et_nat @ A @ B ) )
     => ( ( P @ ( produc8586169260539613262et_nat @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_806_sndE,axiom,
    ! [X2: product_prod_int_int,A: int,B: int,P: int > $o] :
      ( ( X2
        = ( product_Pair_int_int @ A @ B ) )
     => ( ( P @ ( product_snd_int_int @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_807_sndE,axiom,
    ! [X2: produc8923325533196201883nteger,A: code_integer,B: code_integer,P: code_integer > $o] :
      ( ( X2
        = ( produc1086072967326762835nteger @ A @ B ) )
     => ( ( P @ ( produc6174133586879617921nteger @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_808_sndE,axiom,
    ! [X2: produc6575502325842934193n_assn,A: assn,B: assn,P: assn > $o] :
      ( ( X2
        = ( produc118845697133431529n_assn @ A @ B ) )
     => ( ( P @ ( produc2051961928117032727n_assn @ X2 ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_809_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_810_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_811_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_812_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_813_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_814_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_815_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_816_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_817_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_818_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_819_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_820_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_821_option_Osel,axiom,
    ! [X23: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X23 ) )
      = X23 ) ).

% option.sel
thf(fact_822_option_Osel,axiom,
    ! [X23: nat] :
      ( ( the_nat @ ( some_nat @ X23 ) )
      = X23 ) ).

% option.sel
thf(fact_823_option_Osel,axiom,
    ! [X23: num] :
      ( ( the_num @ ( some_num @ X23 ) )
      = X23 ) ).

% option.sel
thf(fact_824_Ex__prod__contract,axiom,
    ! [P: uint32 > uint32 > $o] :
      ( ( ? [A4: uint32,X5: uint32] : ( P @ A4 @ X5 ) )
      = ( ? [Z2: produc827990862158126777uint32] : ( P @ ( produc9004433772639906525uint32 @ Z2 ) @ ( produc1510406741064981791uint32 @ Z2 ) ) ) ) ).

% Ex_prod_contract
thf(fact_825_Ex__prod__contract,axiom,
    ! [P: nat > nat > $o] :
      ( ( ? [A4: nat,X5: nat] : ( P @ A4 @ X5 ) )
      = ( ? [Z2: product_prod_nat_nat] : ( P @ ( product_fst_nat_nat @ Z2 ) @ ( product_snd_nat_nat @ Z2 ) ) ) ) ).

% Ex_prod_contract
thf(fact_826_Ex__prod__contract,axiom,
    ! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
      ( ( ? [A4: heap_e7401611519738050253t_unit,X5: set_nat] : ( P @ A4 @ X5 ) )
      = ( ? [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).

% Ex_prod_contract
thf(fact_827_Ex__prod__contract,axiom,
    ! [P: int > int > $o] :
      ( ( ? [A4: int,X5: int] : ( P @ A4 @ X5 ) )
      = ( ? [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).

% Ex_prod_contract
thf(fact_828_Ex__prod__contract,axiom,
    ! [P: code_integer > code_integer > $o] :
      ( ( ? [A4: code_integer,X5: code_integer] : ( P @ A4 @ X5 ) )
      = ( ? [Z2: produc8923325533196201883nteger] : ( P @ ( produc8508995932063986495nteger @ Z2 ) @ ( produc6174133586879617921nteger @ Z2 ) ) ) ) ).

% Ex_prod_contract
thf(fact_829_Ex__prod__contract,axiom,
    ! [P: assn > assn > $o] :
      ( ( ? [A4: assn,X5: assn] : ( P @ A4 @ X5 ) )
      = ( ? [Z2: produc6575502325842934193n_assn] : ( P @ ( produc9167289414957590229n_assn @ Z2 ) @ ( produc2051961928117032727n_assn @ Z2 ) ) ) ) ).

% Ex_prod_contract
thf(fact_830_All__prod__contract,axiom,
    ! [P: uint32 > uint32 > $o] :
      ( ( ! [A4: uint32,X5: uint32] : ( P @ A4 @ X5 ) )
      = ( ! [Z2: produc827990862158126777uint32] : ( P @ ( produc9004433772639906525uint32 @ Z2 ) @ ( produc1510406741064981791uint32 @ Z2 ) ) ) ) ).

% All_prod_contract
thf(fact_831_All__prod__contract,axiom,
    ! [P: nat > nat > $o] :
      ( ( ! [A4: nat,X5: nat] : ( P @ A4 @ X5 ) )
      = ( ! [Z2: product_prod_nat_nat] : ( P @ ( product_fst_nat_nat @ Z2 ) @ ( product_snd_nat_nat @ Z2 ) ) ) ) ).

% All_prod_contract
thf(fact_832_All__prod__contract,axiom,
    ! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
      ( ( ! [A4: heap_e7401611519738050253t_unit,X5: set_nat] : ( P @ A4 @ X5 ) )
      = ( ! [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).

% All_prod_contract
thf(fact_833_All__prod__contract,axiom,
    ! [P: int > int > $o] :
      ( ( ! [A4: int,X5: int] : ( P @ A4 @ X5 ) )
      = ( ! [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).

% All_prod_contract
thf(fact_834_All__prod__contract,axiom,
    ! [P: code_integer > code_integer > $o] :
      ( ( ! [A4: code_integer,X5: code_integer] : ( P @ A4 @ X5 ) )
      = ( ! [Z2: produc8923325533196201883nteger] : ( P @ ( produc8508995932063986495nteger @ Z2 ) @ ( produc6174133586879617921nteger @ Z2 ) ) ) ) ).

% All_prod_contract
thf(fact_835_All__prod__contract,axiom,
    ! [P: assn > assn > $o] :
      ( ( ! [A4: assn,X5: assn] : ( P @ A4 @ X5 ) )
      = ( ! [Z2: produc6575502325842934193n_assn] : ( P @ ( produc9167289414957590229n_assn @ Z2 ) @ ( produc2051961928117032727n_assn @ Z2 ) ) ) ) ).

% All_prod_contract
thf(fact_836_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_837_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_838_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_839_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_840_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_841_option_Osimps_I5_J,axiom,
    ! [F1: heap_Time_Heap_o,F22: product_prod_nat_nat > heap_Time_Heap_o,X23: product_prod_nat_nat] :
      ( ( case_o1442776274061689234at_nat @ F1 @ F22 @ ( some_P7363390416028606310at_nat @ X23 ) )
      = ( F22 @ X23 ) ) ).

% option.simps(5)
thf(fact_842_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: num > option_num,X23: num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F22 @ ( some_num @ X23 ) )
      = ( F22 @ X23 ) ) ).

% option.simps(5)
thf(fact_843_option_Osimps_I5_J,axiom,
    ! [F1: $o,F22: product_prod_nat_nat > $o,X23: product_prod_nat_nat] :
      ( ( case_o184042715313410164at_nat @ F1 @ F22 @ ( some_P7363390416028606310at_nat @ X23 ) )
      = ( F22 @ X23 ) ) ).

% option.simps(5)
thf(fact_844_option_Osimps_I5_J,axiom,
    ! [F1: num,F22: num > num,X23: num] :
      ( ( case_option_num_num @ F1 @ F22 @ ( some_num @ X23 ) )
      = ( F22 @ X23 ) ) ).

% option.simps(5)
thf(fact_845_option_Osimps_I5_J,axiom,
    ! [F1: int,F22: num > int,X23: num] :
      ( ( case_option_int_num @ F1 @ F22 @ ( some_num @ X23 ) )
      = ( F22 @ X23 ) ) ).

% option.simps(5)
thf(fact_846_option_Osimps_I4_J,axiom,
    ! [F1: heap_Time_Heap_o,F22: product_prod_nat_nat > heap_Time_Heap_o] :
      ( ( case_o1442776274061689234at_nat @ F1 @ F22 @ none_P5556105721700978146at_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_847_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: num > option_num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_848_option_Osimps_I4_J,axiom,
    ! [F1: $o,F22: product_prod_nat_nat > $o] :
      ( ( case_o184042715313410164at_nat @ F1 @ F22 @ none_P5556105721700978146at_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_849_option_Osimps_I4_J,axiom,
    ! [F1: num,F22: num > num] :
      ( ( case_option_num_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_850_option_Osimps_I4_J,axiom,
    ! [F1: int,F22: num > int] :
      ( ( case_option_int_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_851_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_852_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_853_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_854_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_855_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_856_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_857_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_858_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_859_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_860_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_861_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_862_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_863_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_864_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_865_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_866_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_867_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_868_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_869_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_870_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_871_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_872_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_873_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_874_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_875_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_876_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M7: nat,N: nat] :
        ? [K4: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M7 @ K4 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_877_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_878_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_nat @ B @ ( suc @ A ) ) )
      = ( B = A ) ) ).

% nat_in_between_eq(2)
thf(fact_879_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ B @ ( suc @ A ) ) )
      = ( B
        = ( suc @ A ) ) ) ).

% nat_in_between_eq(1)
thf(fact_880_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_881_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_882_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_883_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_884_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_885_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_886_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_887_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_888_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_889_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_890_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_891_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_892_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_893_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_894_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_895_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M2: nat,N4: nat] :
          ( ( ord_less_nat @ M2 @ N4 )
         => ( ord_less_nat @ ( F @ M2 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_896_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_897_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_898_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_899_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_900_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_901_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_902_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_903_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_904_mlex__snd__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,B3: nat,N3: nat] :
      ( ( A = A5 )
     => ( ( ord_less_nat @ B @ B3 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N3 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N3 ) @ B3 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_905_mlex__fst__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,N3: nat,B3: nat] :
      ( ( ord_less_nat @ A @ A5 )
     => ( ( ord_less_nat @ B @ N3 )
       => ( ( ord_less_nat @ B3 @ N3 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N3 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N3 ) @ B3 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_906_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N3: nat] :
      ( ( ord_less_nat @ A @ A2 )
     => ( ( ord_less_nat @ B @ N3 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N3 ) @ B ) @ ( times_times_nat @ A2 @ N3 ) ) ) ) ).

% mlex_bound
thf(fact_907_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_908_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_909_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_910_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_911_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_912_obtain__list__from__elements,axiom,
    ! [N2: nat,P: vEBT_VEBT > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: vEBT_VEBT] : ( P @ Li @ I3 ) )
     => ~ ! [L3: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P @ ( nth_VEBT_VEBT @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_913_obtain__list__from__elements,axiom,
    ! [N2: nat,P: vEBT_VEBTi > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: vEBT_VEBTi] : ( P @ Li @ I3 ) )
     => ~ ! [L3: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P @ ( nth_VEBT_VEBTi @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_914_obtain__list__from__elements,axiom,
    ! [N2: nat,P: real > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: real] : ( P @ Li @ I3 ) )
     => ~ ! [L3: list_real] :
            ( ( ( size_size_list_real @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P @ ( nth_real @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_915_obtain__list__from__elements,axiom,
    ! [N2: nat,P: $o > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: $o] : ( P @ Li @ I3 ) )
     => ~ ! [L3: list_o] :
            ( ( ( size_size_list_o @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P @ ( nth_o @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_916_obtain__list__from__elements,axiom,
    ! [N2: nat,P: nat > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: nat] : ( P @ Li @ I3 ) )
     => ~ ! [L3: list_nat] :
            ( ( ( size_size_list_nat @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P @ ( nth_nat @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_917_obtain__list__from__elements,axiom,
    ! [N2: nat,P: int > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: int] : ( P @ Li @ I3 ) )
     => ~ ! [L3: list_int] :
            ( ( ( size_size_list_int @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P @ ( nth_int @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_918_mlex__leI,axiom,
    ! [A: nat,A5: nat,B: nat,B3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ A @ A5 )
     => ( ( ord_less_eq_nat @ B @ B3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N3 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N3 ) @ B3 ) ) ) ) ).

% mlex_leI
thf(fact_919_div__mult__le,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ A ) ).

% div_mult_le
thf(fact_920_option_Oexhaust__sel,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( Option
        = ( some_P2407035485129114418it_nat @ ( the_Pr3501439614016493281it_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_921_option_Oexhaust__sel,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( Option
        = ( some_P468703482102919278it_nat @ ( the_Pr5838048819577852031it_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_922_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_923_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_924_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_925_option_Ocase__eq__if,axiom,
    ( case_o1442776274061689234at_nat
    = ( ^ [F12: 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 ) @ F12 @ ( F23 @ ( the_Pr8591224930841456533at_nat @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_926_option_Ocase__eq__if,axiom,
    ( case_o6005452278849405969um_num
    = ( ^ [F12: option_num,F23: num > option_num,Option3: option_num] : ( if_option_num @ ( Option3 = none_num ) @ F12 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_927_option_Ocase__eq__if,axiom,
    ( case_o184042715313410164at_nat
    = ( ^ [F12: $o,F23: product_prod_nat_nat > $o,Option3: option4927543243414619207at_nat] :
          ( ( ( Option3 = none_P5556105721700978146at_nat )
           => F12 )
          & ( ( Option3 != none_P5556105721700978146at_nat )
           => ( F23 @ ( the_Pr8591224930841456533at_nat @ Option3 ) ) ) ) ) ) ).

% option.case_eq_if
thf(fact_928_option_Ocase__eq__if,axiom,
    ( case_option_num_num
    = ( ^ [F12: num,F23: num > num,Option3: option_num] : ( if_num @ ( Option3 = none_num ) @ F12 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_929_option_Ocase__eq__if,axiom,
    ( case_option_int_num
    = ( ^ [F12: int,F23: num > int,Option3: option_num] : ( if_int @ ( Option3 = none_num ) @ F12 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_930_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 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs )
             => ( ( ord_less_nat @ X @ Z2 )
               => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_option_Osplit__sel,axiom,
    ! [P: heap_Time_Heap_o > $o,F1: heap_Time_Heap_o,F22: product_prod_nat_nat > heap_Time_Heap_o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o1442776274061689234at_nat @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_P5556105721700978146at_nat )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
         => ( P @ ( F22 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_939_option_Osplit__sel,axiom,
    ! [P: option_num > $o,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( P @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_940_option_Osplit__sel,axiom,
    ! [P: $o > $o,F1: $o,F22: product_prod_nat_nat > $o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o184042715313410164at_nat @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_P5556105721700978146at_nat )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
         => ( P @ ( F22 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_941_option_Osplit__sel,axiom,
    ! [P: num > $o,F1: num,F22: num > num,Option: option_num] :
      ( ( P @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_942_option_Osplit__sel,axiom,
    ! [P: int > $o,F1: int,F22: num > int,Option: option_num] :
      ( ( P @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_943_option_Osplit__sel__asm,axiom,
    ! [P: heap_Time_Heap_o > $o,F1: heap_Time_Heap_o,F22: product_prod_nat_nat > heap_Time_Heap_o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o1442776274061689234at_nat @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_P5556105721700978146at_nat )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_944_option_Osplit__sel__asm,axiom,
    ! [P: option_num > $o,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( P @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_945_option_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F1: $o,F22: product_prod_nat_nat > $o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o184042715313410164at_nat @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_P5556105721700978146at_nat )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_946_option_Osplit__sel__asm,axiom,
    ! [P: num > $o,F1: num,F22: num > num,Option: option_num] :
      ( ( P @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_947_option_Osplit__sel__asm,axiom,
    ! [P: int > $o,F1: int,F22: num > int,Option: option_num] :
      ( ( P @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_948_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_949_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_950_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_951_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_952_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_953_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( 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_954_prod_Ocollapse,axiom,
    ! [Prod: produc827990862158126777uint32] :
      ( ( produc1400373151660368625uint32 @ ( produc9004433772639906525uint32 @ Prod ) @ ( produc1510406741064981791uint32 @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_955_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_956_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_957_prod_Ocollapse,axiom,
    ! [Prod: produc8923325533196201883nteger] :
      ( ( produc1086072967326762835nteger @ ( produc8508995932063986495nteger @ Prod ) @ ( produc6174133586879617921nteger @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_958_prod_Ocollapse,axiom,
    ! [Prod: produc6575502325842934193n_assn] :
      ( ( produc118845697133431529n_assn @ ( produc9167289414957590229n_assn @ Prod ) @ ( produc2051961928117032727n_assn @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_959_prod_Ocollapse,axiom,
    ! [Prod: produc3658429121746597890et_nat] :
      ( ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_960_prod_Ocollapse,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_961_prod_Ocollapse,axiom,
    ! [Prod: produc3925858234332021118et_nat] :
      ( ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_962_prod_Ocollapse,axiom,
    ! [Prod: produc1908205239877642774nteger] :
      ( ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_963_prod_Ocollapse,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_964_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_965_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_966_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_967_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_968_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_969_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_970_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_971_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_972_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% two_powr_height_bound_deg
thf(fact_973_setprop,axiom,
    ! [T: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ T @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% setprop
thf(fact_974_norm__assertion__simps_I17_J,axiom,
    ! [R2: assn,Q: list_VEBT_VEBTi > assn] :
      ( ( times_times_assn @ R2 @ ( ex_ass463751140784270563_VEBTi @ Q ) )
      = ( ex_ass463751140784270563_VEBTi
        @ ^ [X: list_VEBT_VEBTi] : ( times_times_assn @ R2 @ ( Q @ X ) ) ) ) ).

% norm_assertion_simps(17)
thf(fact_975_norm__assertion__simps_I16_J,axiom,
    ! [Q: list_VEBT_VEBTi > assn,R2: assn] :
      ( ( times_times_assn @ ( ex_ass463751140784270563_VEBTi @ Q ) @ R2 )
      = ( ex_ass463751140784270563_VEBTi
        @ ^ [X: list_VEBT_VEBTi] : ( times_times_assn @ ( Q @ X ) @ R2 ) ) ) ).

% norm_assertion_simps(16)
thf(fact_976_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,Va: nat,TreeList: list_VEBT_VEBT,Summary: 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 @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_977_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,Va: nat,TreeList: list_VEBT_VEBT,Summary: 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 @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_978_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_old_Oprod_Oinject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_987_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_988_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_989_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_990_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_991_prod_Oinject,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X23: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y23: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X23 )
        = ( produc8603105652947943368nteger @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_992_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X23: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y23: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ X1 @ X23 )
        = ( produc5001842942810119800et_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_993_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X23: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y23: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ X1 @ X23 )
        = ( produc2245416461498447860et_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_994_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X23: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y23: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X23 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_995_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X23: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y23: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X23 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_996_real__divide__square__eq,axiom,
    ! [R3: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R3 @ A ) @ ( times_times_real @ R3 @ R3 ) )
      = ( divide_divide_real @ A @ R3 ) ) ).

% real_divide_square_eq
thf(fact_997_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% height_compose_summary
thf(fact_998_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_compose_child
thf(fact_999_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_1000_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% delete_bound_height'
thf(fact_1001_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_1002_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_1003_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_1004_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_1005_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option = none_P1551326421579882414it_nat )
      = ( case_o535201446637900608it_nat @ $true
        @ ^ [Uu3: produc120671012495760973it_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_1006_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option = none_P7668321371905463026it_nat )
      = ( case_o1358941076187788256it_nat @ $true
        @ ^ [Uu3: produc8047831477865546771it_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_1007_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
      = ( case_option_o_num @ $true
        @ ^ [Uu3: num] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_1008_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_nat] :
      ( ( Option = none_nat )
      = ( case_option_o_nat @ $true
        @ ^ [Uu3: nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_1009_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option = none_P5556105721700978146at_nat )
      = ( case_o184042715313410164at_nat @ $true
        @ ^ [Uu3: product_prod_nat_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_1010_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
      = ( case_o535201446637900608it_nat @ $false
        @ ^ [Uu3: produc120671012495760973it_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_1011_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
      = ( case_o1358941076187788256it_nat @ $false
        @ ^ [Uu3: produc8047831477865546771it_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_1012_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
      = ( case_option_o_num @ $false
        @ ^ [Uu3: num] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_1013_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
      = ( case_option_o_nat @ $false
        @ ^ [Uu3: nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_1014_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
      = ( case_o184042715313410164at_nat @ $false
        @ ^ [Uu3: product_prod_nat_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_1015_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_1016_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_1017_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d2 @ T @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% pred_bound_height'
thf(fact_1018_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c2 @ T @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% succ'_bound_height
thf(fact_1019_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_1020_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_1021_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_1022_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_1023_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_1024_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ L ) )
           => ( P @ ( nth_VEBT_VEBTi @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1025_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ L ) )
           => ( P @ ( nth_VEBT_VEBT @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1026_all__set__conv__nth,axiom,
    ! [L: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ! [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6829681357464350627n_assn @ L ) )
           => ( P @ ( nth_Pr1769885009046257848n_assn @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1027_all__set__conv__nth,axiom,
    ! [L: list_real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ L ) )
           => ( P @ ( nth_real @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1028_all__set__conv__nth,axiom,
    ! [L: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ L ) )
           => ( P @ ( nth_o @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1029_all__set__conv__nth,axiom,
    ! [L: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( P @ ( nth_nat @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1030_all__set__conv__nth,axiom,
    ! [L: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ L ) )
           => ( P @ ( nth_int @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1031_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_1032_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_1033_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_1034_prod__induct4,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X2: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D3: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D3 ) ) ) )
     => ( P @ X2 ) ) ).

% prod_induct4
thf(fact_1035_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X2: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_1036_prod__induct3,axiom,
    ! [P: produc3925858234332021118et_nat > $o,X2: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_1037_prod__induct3,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X2: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_1038_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X2: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_1039_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X2: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_1040_prod__cases4,axiom,
    ! [Y2: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D3: set_nat] :
          ( Y2
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D3 ) ) ) ) ).

% prod_cases4
thf(fact_1041_prod__cases3,axiom,
    ! [Y2: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C2: code_integer] :
          ( Y2
         != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1042_prod__cases3,axiom,
    ! [Y2: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] :
          ( Y2
         != ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1043_prod__cases3,axiom,
    ! [Y2: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] :
          ( Y2
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1044_prod__cases3,axiom,
    ! [Y2: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C2: int] :
          ( Y2
         != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1045_prod__cases3,axiom,
    ! [Y2: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B2: int,C2: int] :
          ( Y2
         != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1046_Pair__inject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_1047_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_1048_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_1049_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_1050_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_1051_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1052_prod__cases,axiom,
    ! [P: produc3925858234332021118et_nat > $o,P4: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1053_prod__cases,axiom,
    ! [P: produc2732055786443039994et_nat > $o,P4: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1054_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1055_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1056_surj__pair,axiom,
    ! [P4: produc1908205239877642774nteger] :
    ? [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
      ( P4
      = ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_1057_surj__pair,axiom,
    ! [P4: produc3925858234332021118et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
      ( P4
      = ( produc5001842942810119800et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_1058_surj__pair,axiom,
    ! [P4: produc2732055786443039994et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
      ( P4
      = ( produc2245416461498447860et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_1059_surj__pair,axiom,
    ! [P4: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P4
      = ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_1060_surj__pair,axiom,
    ! [P4: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P4
      = ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_1061_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( Y2
         != ( produc8603105652947943368nteger @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_1062_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( Y2
         != ( produc5001842942810119800et_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_1063_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
          ( Y2
         != ( produc2245416461498447860et_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_1064_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( Y2
         != ( produc5700946648718959541nt_int @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_1065_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( Y2
         != ( produc4305682042979456191nt_int @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_1066_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% star_assoc
thf(fact_1067_star__aci_I2_J,axiom,
    ( times_times_assn
    = ( ^ [A4: assn,B4: assn] : ( times_times_assn @ B4 @ A4 ) ) ) ).

% star_aci(2)
thf(fact_1068_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ A @ ( times_times_assn @ B @ C ) )
      = ( times_times_assn @ B @ ( times_times_assn @ A @ C ) ) ) ).

% star_aci(3)
thf(fact_1069_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ ( times_times_assn @ A @ C ) @ B ) ) ).

% assn_aci(10)
thf(fact_1070_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_1071_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_1072_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_1073_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_1074_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ( hoare_8945653483474564448t_unit @ P @ C @ Q )
     => ( hoare_8945653483474564448t_unit @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_1075_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ord_less_eq_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_node
thf(fact_1076_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_1077_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_1078_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_1079_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_1080_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_1081_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_1082_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_1083_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_1084_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_1085_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_1086_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_1087_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_1088_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1089_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1090_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1091_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1092_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1093_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1094_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1095_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1096_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1097_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1098_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1099_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1100_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1101_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1102_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1103_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1104_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1105_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1106_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1107_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1108_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1109_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1110_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1111_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1112_fst__conv,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X23: produc8923325533196201883nteger] :
      ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1113_fst__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X23: produc3658429121746597890et_nat] :
      ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1114_fst__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X23: produc3925858234332021118et_nat] :
      ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1115_fst__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X23: product_prod_int_int] :
      ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1116_fst__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X23: product_prod_int_int] :
      ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1117_fst__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X23: set_nat] :
      ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1118_fst__conv,axiom,
    ! [X1: assn,X23: assn] :
      ( ( produc9167289414957590229n_assn @ ( produc118845697133431529n_assn @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1119_fst__conv,axiom,
    ! [X1: uint32,X23: uint32] :
      ( ( produc9004433772639906525uint32 @ ( produc1400373151660368625uint32 @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1120_fst__conv,axiom,
    ! [X1: nat,X23: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1121_fst__conv,axiom,
    ! [X1: int,X23: int] :
      ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_1122_fst__eqD,axiom,
    ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1123_fst__eqD,axiom,
    ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,A: produc3658429121746597890et_nat > $o] :
      ( ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1124_fst__eqD,axiom,
    ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,A: produc3658429121746597890et_nat > $o] :
      ( ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1125_fst__eqD,axiom,
    ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int,A: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1126_fst__eqD,axiom,
    ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int,A: int > option6357759511663192854e_term] :
      ( ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1127_fst__eqD,axiom,
    ! [X2: heap_e7401611519738050253t_unit,Y2: set_nat,A: heap_e7401611519738050253t_unit] :
      ( ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1128_fst__eqD,axiom,
    ! [X2: assn,Y2: assn,A: assn] :
      ( ( ( produc9167289414957590229n_assn @ ( produc118845697133431529n_assn @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1129_fst__eqD,axiom,
    ! [X2: uint32,Y2: uint32,A: uint32] :
      ( ( ( produc9004433772639906525uint32 @ ( produc1400373151660368625uint32 @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1130_fst__eqD,axiom,
    ! [X2: nat,Y2: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1131_fst__eqD,axiom,
    ! [X2: int,Y2: int,A: int] :
      ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X2 @ Y2 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_1132_snd__conv,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X23: produc8923325533196201883nteger] :
      ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1133_snd__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X23: produc3658429121746597890et_nat] :
      ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1134_snd__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X23: produc3925858234332021118et_nat] :
      ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1135_snd__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X23: product_prod_int_int] :
      ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1136_snd__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X23: product_prod_int_int] :
      ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1137_snd__conv,axiom,
    ! [X1: nat,X23: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1138_snd__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X23: set_nat] :
      ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1139_snd__conv,axiom,
    ! [X1: int,X23: int] :
      ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1140_snd__conv,axiom,
    ! [X1: code_integer,X23: code_integer] :
      ( ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1141_snd__conv,axiom,
    ! [X1: assn,X23: assn] :
      ( ( produc2051961928117032727n_assn @ ( produc118845697133431529n_assn @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_1142_snd__eqD,axiom,
    ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
      ( ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1143_snd__eqD,axiom,
    ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,A: produc3658429121746597890et_nat] :
      ( ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1144_snd__eqD,axiom,
    ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,A: produc3925858234332021118et_nat] :
      ( ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1145_snd__eqD,axiom,
    ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1146_snd__eqD,axiom,
    ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1147_snd__eqD,axiom,
    ! [X2: nat,Y2: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1148_snd__eqD,axiom,
    ! [X2: heap_e7401611519738050253t_unit,Y2: set_nat,A: set_nat] :
      ( ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1149_snd__eqD,axiom,
    ! [X2: int,Y2: int,A: int] :
      ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1150_snd__eqD,axiom,
    ! [X2: code_integer,Y2: code_integer,A: code_integer] :
      ( ( ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1151_snd__eqD,axiom,
    ! [X2: assn,Y2: assn,A: assn] :
      ( ( ( produc2051961928117032727n_assn @ ( produc118845697133431529n_assn @ X2 @ Y2 ) )
        = A )
     => ( Y2 = A ) ) ).

% snd_eqD
thf(fact_1152_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc827990862158126777uint32,Z4: produc827990862158126777uint32] : Y5 = Z4 )
    = ( ^ [S4: produc827990862158126777uint32,T2: produc827990862158126777uint32] :
          ( ( ( produc9004433772639906525uint32 @ S4 )
            = ( produc9004433772639906525uint32 @ T2 ) )
          & ( ( produc1510406741064981791uint32 @ S4 )
            = ( produc1510406741064981791uint32 @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_1153_prod__eq__iff,axiom,
    ( ( ^ [Y5: product_prod_nat_nat,Z4: product_prod_nat_nat] : Y5 = Z4 )
    = ( ^ [S4: product_prod_nat_nat,T2: product_prod_nat_nat] :
          ( ( ( product_fst_nat_nat @ S4 )
            = ( product_fst_nat_nat @ T2 ) )
          & ( ( product_snd_nat_nat @ S4 )
            = ( product_snd_nat_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_1154_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc3658429121746597890et_nat,Z4: produc3658429121746597890et_nat] : Y5 = Z4 )
    = ( ^ [S4: produc3658429121746597890et_nat,T2: produc3658429121746597890et_nat] :
          ( ( ( produc1824681642469235216et_nat @ S4 )
            = ( produc1824681642469235216et_nat @ T2 ) )
          & ( ( produc8586169260539613262et_nat @ S4 )
            = ( produc8586169260539613262et_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_1155_prod__eq__iff,axiom,
    ( ( ^ [Y5: product_prod_int_int,Z4: product_prod_int_int] : Y5 = Z4 )
    = ( ^ [S4: product_prod_int_int,T2: product_prod_int_int] :
          ( ( ( product_fst_int_int @ S4 )
            = ( product_fst_int_int @ T2 ) )
          & ( ( product_snd_int_int @ S4 )
            = ( product_snd_int_int @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_1156_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc8923325533196201883nteger,Z4: produc8923325533196201883nteger] : Y5 = Z4 )
    = ( ^ [S4: produc8923325533196201883nteger,T2: produc8923325533196201883nteger] :
          ( ( ( produc8508995932063986495nteger @ S4 )
            = ( produc8508995932063986495nteger @ T2 ) )
          & ( ( produc6174133586879617921nteger @ S4 )
            = ( produc6174133586879617921nteger @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_1157_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc6575502325842934193n_assn,Z4: produc6575502325842934193n_assn] : Y5 = Z4 )
    = ( ^ [S4: produc6575502325842934193n_assn,T2: produc6575502325842934193n_assn] :
          ( ( ( produc9167289414957590229n_assn @ S4 )
            = ( produc9167289414957590229n_assn @ T2 ) )
          & ( ( produc2051961928117032727n_assn @ S4 )
            = ( produc2051961928117032727n_assn @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_1158_prod__eqI,axiom,
    ! [P4: produc827990862158126777uint32,Q2: produc827990862158126777uint32] :
      ( ( ( produc9004433772639906525uint32 @ P4 )
        = ( produc9004433772639906525uint32 @ Q2 ) )
     => ( ( ( produc1510406741064981791uint32 @ P4 )
          = ( produc1510406741064981791uint32 @ Q2 ) )
       => ( P4 = Q2 ) ) ) ).

% prod_eqI
thf(fact_1159_prod__eqI,axiom,
    ! [P4: product_prod_nat_nat,Q2: product_prod_nat_nat] :
      ( ( ( product_fst_nat_nat @ P4 )
        = ( product_fst_nat_nat @ Q2 ) )
     => ( ( ( product_snd_nat_nat @ P4 )
          = ( product_snd_nat_nat @ Q2 ) )
       => ( P4 = Q2 ) ) ) ).

% prod_eqI
thf(fact_1160_prod__eqI,axiom,
    ! [P4: produc3658429121746597890et_nat,Q2: produc3658429121746597890et_nat] :
      ( ( ( produc1824681642469235216et_nat @ P4 )
        = ( produc1824681642469235216et_nat @ Q2 ) )
     => ( ( ( produc8586169260539613262et_nat @ P4 )
          = ( produc8586169260539613262et_nat @ Q2 ) )
       => ( P4 = Q2 ) ) ) ).

% prod_eqI
thf(fact_1161_prod__eqI,axiom,
    ! [P4: product_prod_int_int,Q2: product_prod_int_int] :
      ( ( ( product_fst_int_int @ P4 )
        = ( product_fst_int_int @ Q2 ) )
     => ( ( ( product_snd_int_int @ P4 )
          = ( product_snd_int_int @ Q2 ) )
       => ( P4 = Q2 ) ) ) ).

% prod_eqI
thf(fact_1162_prod__eqI,axiom,
    ! [P4: produc8923325533196201883nteger,Q2: produc8923325533196201883nteger] :
      ( ( ( produc8508995932063986495nteger @ P4 )
        = ( produc8508995932063986495nteger @ Q2 ) )
     => ( ( ( produc6174133586879617921nteger @ P4 )
          = ( produc6174133586879617921nteger @ Q2 ) )
       => ( P4 = Q2 ) ) ) ).

% prod_eqI
thf(fact_1163_prod__eqI,axiom,
    ! [P4: produc6575502325842934193n_assn,Q2: produc6575502325842934193n_assn] :
      ( ( ( produc9167289414957590229n_assn @ P4 )
        = ( produc9167289414957590229n_assn @ Q2 ) )
     => ( ( ( produc2051961928117032727n_assn @ P4 )
          = ( produc2051961928117032727n_assn @ Q2 ) )
       => ( P4 = Q2 ) ) ) ).

% prod_eqI
thf(fact_1164_prod_Oexpand,axiom,
    ! [Prod: produc827990862158126777uint32,Prod2: produc827990862158126777uint32] :
      ( ( ( ( produc9004433772639906525uint32 @ Prod )
          = ( produc9004433772639906525uint32 @ Prod2 ) )
        & ( ( produc1510406741064981791uint32 @ Prod )
          = ( produc1510406741064981791uint32 @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_1165_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_1166_prod_Oexpand,axiom,
    ! [Prod: produc3658429121746597890et_nat,Prod2: produc3658429121746597890et_nat] :
      ( ( ( ( produc1824681642469235216et_nat @ Prod )
          = ( produc1824681642469235216et_nat @ Prod2 ) )
        & ( ( produc8586169260539613262et_nat @ Prod )
          = ( produc8586169260539613262et_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_1167_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_1168_prod_Oexpand,axiom,
    ! [Prod: produc8923325533196201883nteger,Prod2: produc8923325533196201883nteger] :
      ( ( ( ( produc8508995932063986495nteger @ Prod )
          = ( produc8508995932063986495nteger @ Prod2 ) )
        & ( ( produc6174133586879617921nteger @ Prod )
          = ( produc6174133586879617921nteger @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_1169_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_1170_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% norm_assertion_simps(1)
thf(fact_1171_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% norm_assertion_simps(2)
thf(fact_1172_lambda__one,axiom,
    ( ( ^ [X: word_N3645301735248828278l_num1] : X )
    = ( times_7065122842183080059l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% lambda_one
thf(fact_1173_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_1174_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_1175_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_1176_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_1177_frame__rule__left,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,R2: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ R2 @ P ) @ C
        @ ^ [X: option_nat] : ( times_times_assn @ R2 @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_1178_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,R2: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ R2 @ P ) @ C
        @ ^ [X: nat] : ( times_times_assn @ R2 @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_1179_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,R2: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ R2 @ P ) @ C
        @ ^ [X: $o] : ( times_times_assn @ R2 @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_1180_frame__rule__left,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R2: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ R2 @ P ) @ C
        @ ^ [X: vEBT_VEBTi] : ( times_times_assn @ R2 @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_1181_frame__rule__left,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn,R2: assn] :
      ( ( hoare_8945653483474564448t_unit @ P @ C @ Q )
     => ( hoare_8945653483474564448t_unit @ ( times_times_assn @ R2 @ P ) @ C
        @ ^ [X: product_unit] : ( times_times_assn @ R2 @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_1182_add__mono1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).

% add_mono1
thf(fact_1183_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_1184_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_1185_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_1186_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_1187_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_1188_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_1189_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_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_1202_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ ( ord_less_real @ A @ B )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1203_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1204_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1205_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1206_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_1207_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_1208_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_1209_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1210_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1211_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1212_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1213_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1214_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1215_surjective__pairing,axiom,
    ! [T: produc827990862158126777uint32] :
      ( T
      = ( produc1400373151660368625uint32 @ ( produc9004433772639906525uint32 @ T ) @ ( produc1510406741064981791uint32 @ T ) ) ) ).

% surjective_pairing
thf(fact_1216_surjective__pairing,axiom,
    ! [T: product_prod_nat_nat] :
      ( T
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_1217_surjective__pairing,axiom,
    ! [T: product_prod_int_int] :
      ( T
      = ( product_Pair_int_int @ ( product_fst_int_int @ T ) @ ( product_snd_int_int @ T ) ) ) ).

% surjective_pairing
thf(fact_1218_surjective__pairing,axiom,
    ! [T: produc8923325533196201883nteger] :
      ( T
      = ( produc1086072967326762835nteger @ ( produc8508995932063986495nteger @ T ) @ ( produc6174133586879617921nteger @ T ) ) ) ).

% surjective_pairing
thf(fact_1219_surjective__pairing,axiom,
    ! [T: produc6575502325842934193n_assn] :
      ( T
      = ( produc118845697133431529n_assn @ ( produc9167289414957590229n_assn @ T ) @ ( produc2051961928117032727n_assn @ T ) ) ) ).

% surjective_pairing
thf(fact_1220_surjective__pairing,axiom,
    ! [T: produc3658429121746597890et_nat] :
      ( T
      = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ T ) @ ( produc8586169260539613262et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_1221_surjective__pairing,axiom,
    ! [T: produc7773217078559923341nt_int] :
      ( T
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ T ) @ ( produc3162348030201620241nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_1222_surjective__pairing,axiom,
    ! [T: produc3925858234332021118et_nat] :
      ( T
      = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ T ) @ ( produc4011572625026189258et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_1223_surjective__pairing,axiom,
    ! [T: produc1908205239877642774nteger] :
      ( T
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ T ) @ ( produc7856867400915047194nteger @ T ) ) ) ).

% surjective_pairing
thf(fact_1224_surjective__pairing,axiom,
    ! [T: produc2285326912895808259nt_int] :
      ( T
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ T ) @ ( produc7328097813583171335nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_1225_prod_Oexhaust__sel,axiom,
    ! [Prod: produc827990862158126777uint32] :
      ( Prod
      = ( produc1400373151660368625uint32 @ ( produc9004433772639906525uint32 @ Prod ) @ ( produc1510406741064981791uint32 @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1226_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_1227_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_1228_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8923325533196201883nteger] :
      ( Prod
      = ( produc1086072967326762835nteger @ ( produc8508995932063986495nteger @ Prod ) @ ( produc6174133586879617921nteger @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1229_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6575502325842934193n_assn] :
      ( Prod
      = ( produc118845697133431529n_assn @ ( produc9167289414957590229n_assn @ Prod ) @ ( produc2051961928117032727n_assn @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1230_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3658429121746597890et_nat] :
      ( Prod
      = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1231_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( Prod
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1232_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3925858234332021118et_nat] :
      ( Prod
      = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1233_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1908205239877642774nteger] :
      ( Prod
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1234_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( Prod
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1235_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1236_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1237_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1238_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1239_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1240_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1241_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_1242_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_1243_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_1244_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_1245_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_1246_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_1247_square__diff__one__factored,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ X2 @ X2 ) @ one_on7727431528512463931l_num1 )
      = ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ one_on7727431528512463931l_num1 ) @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) ) ) ).

% square_diff_one_factored
thf(fact_1248_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_1249_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_1250_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_1251_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_1252_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_1253_real__average__minus__first,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_1254_real__average__minus__second,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_1255_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_1256_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_1257_less__eq__option__Some__None,axiom,
    ! [X2: nat] :
      ~ ( ord_le5914376470875661696on_nat @ ( some_nat @ X2 ) @ none_nat ) ).

% less_eq_option_Some_None
thf(fact_1258_less__eq__option__Some__None,axiom,
    ! [X2: num] :
      ~ ( ord_le6622620407824499402on_num @ ( some_num @ X2 ) @ none_num ) ).

% less_eq_option_Some_None
thf(fact_1259_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_1260_less__eq__option__Some,axiom,
    ! [X2: set_int,Y2: set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ X2 ) @ ( some_set_int @ Y2 ) )
      = ( ord_less_eq_set_int @ X2 @ Y2 ) ) ).

% less_eq_option_Some
thf(fact_1261_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_1262_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_1263_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_1264_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_1265_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_1266_less__option__None,axiom,
    ! [X2: option_num] :
      ~ ( ord_less_option_num @ X2 @ none_num ) ).

% less_option_None
thf(fact_1267_less__option__None,axiom,
    ! [X2: option_nat] :
      ~ ( ord_less_option_nat @ X2 @ none_nat ) ).

% less_option_None
thf(fact_1268_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_1269_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_1270_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_1271_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_1272_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_1273_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_1274_less__option__None__Some__code,axiom,
    ! [X2: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X2 ) ) ).

% less_option_None_Some_code
thf(fact_1275_less__option__None__Some__code,axiom,
    ! [X2: num] : ( ord_less_option_num @ none_num @ ( some_num @ X2 ) ) ).

% less_option_None_Some_code
thf(fact_1276_less__eq__option__None__code,axiom,
    ! [X2: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X2 ) ).

% less_eq_option_None_code
thf(fact_1277_less__eq__option__None__code,axiom,
    ! [X2: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X2 ) ).

% less_eq_option_None_code
thf(fact_1278_less__option__def,axiom,
    ( ord_less_option_real
    = ( ^ [X: option_real] :
          ( case_option_o_real @ $false
          @ ^ [Y: real] :
              ( case_option_o_real @ $true
              @ ^ [Z2: real] : ( ord_less_real @ Z2 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_1279_less__option__def,axiom,
    ( ord_less_option_rat
    = ( ^ [X: option_rat] :
          ( case_option_o_rat @ $false
          @ ^ [Y: rat] :
              ( case_option_o_rat @ $true
              @ ^ [Z2: rat] : ( ord_less_rat @ Z2 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_1280_less__option__def,axiom,
    ( ord_less_option_num
    = ( ^ [X: option_num] :
          ( case_option_o_num @ $false
          @ ^ [Y: num] :
              ( case_option_o_num @ $true
              @ ^ [Z2: num] : ( ord_less_num @ Z2 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_1281_less__option__def,axiom,
    ( ord_less_option_nat
    = ( ^ [X: option_nat] :
          ( case_option_o_nat @ $false
          @ ^ [Y: nat] :
              ( case_option_o_nat @ $true
              @ ^ [Z2: nat] : ( ord_less_nat @ Z2 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_1282_less__option__def,axiom,
    ( ord_less_option_int
    = ( ^ [X: option_int] :
          ( case_option_o_int @ $false
          @ ^ [Y: int] :
              ( case_option_o_int @ $true
              @ ^ [Z2: int] : ( ord_less_int @ Z2 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_1283_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_1284_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_1285_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_1286_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_1287_less__option__None__Some,axiom,
    ! [X2: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X2 ) ) ).

% less_option_None_Some
thf(fact_1288_less__option__None__Some,axiom,
    ! [X2: num] : ( ord_less_option_num @ none_num @ ( some_num @ X2 ) ) ).

% less_option_None_Some
thf(fact_1289_less__option__None__is__Some,axiom,
    ! [X2: option_nat] :
      ( ( ord_less_option_nat @ none_nat @ X2 )
     => ? [Z3: nat] :
          ( X2
          = ( some_nat @ Z3 ) ) ) ).

% less_option_None_is_Some
thf(fact_1290_less__option__None__is__Some,axiom,
    ! [X2: option_num] :
      ( ( ord_less_option_num @ none_num @ X2 )
     => ? [Z3: num] :
          ( X2
          = ( some_num @ Z3 ) ) ) ).

% less_option_None_is_Some
thf(fact_1291_less__eq__option__None,axiom,
    ! [X2: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X2 ) ).

% less_eq_option_None
thf(fact_1292_less__eq__option__None,axiom,
    ! [X2: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X2 ) ).

% less_eq_option_None
thf(fact_1293_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_1294_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_1295_less__eq__option__def,axiom,
    ( ord_le353528952715127954et_int
    = ( ^ [X: option_set_int,Y: option_set_int] :
          ( case_o223999843215110191et_int @ $true
          @ ^ [Z2: set_int] : ( case_o223999843215110191et_int @ $false @ ( ord_less_eq_set_int @ Z2 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_1296_less__eq__option__def,axiom,
    ( ord_le2406147912482264968on_rat
    = ( ^ [X: option_rat,Y: option_rat] :
          ( case_option_o_rat @ $true
          @ ^ [Z2: rat] : ( case_option_o_rat @ $false @ ( ord_less_eq_rat @ Z2 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_1297_less__eq__option__def,axiom,
    ( ord_le6622620407824499402on_num
    = ( ^ [X: option_num,Y: option_num] :
          ( case_option_o_num @ $true
          @ ^ [Z2: num] : ( case_option_o_num @ $false @ ( ord_less_eq_num @ Z2 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_1298_less__eq__option__def,axiom,
    ( ord_le5914376470875661696on_nat
    = ( ^ [X: option_nat,Y: option_nat] :
          ( case_option_o_nat @ $true
          @ ^ [Z2: nat] : ( case_option_o_nat @ $false @ ( ord_less_eq_nat @ Z2 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_1299_less__eq__option__def,axiom,
    ( ord_le1736525451366464988on_int
    = ( ^ [X: option_int,Y: option_int] :
          ( case_option_o_int @ $true
          @ ^ [Z2: int] : ( case_option_o_int @ $false @ ( ord_less_eq_int @ Z2 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_1300_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_1301_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1302_discrete,axiom,
    ( ord_less_int
    = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1303_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_1304_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_1305_divmod__step__eq,axiom,
    ! [L: num,R3: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_1306_divmod__step__eq,axiom,
    ! [L: num,R3: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R3 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R3 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_1307_divmod__step__eq,axiom,
    ! [L: num,R3: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_1308_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( 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_1309_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND_o @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_memberi
thf(fact_1310_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_1311_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_1312_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_1313_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N2: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_P1551326421579882414it_nat = none_P1551326421579882414it_nat )
                & ( N2 = N2 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N2 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N2 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_1314_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N2: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_P7668321371905463026it_nat = none_P7668321371905463026it_nat )
                & ( N2 = N2 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N2 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N2 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_1315_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N2: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                & ( N2 = N2 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N2 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N2 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_1316_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N2: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_num = none_num )
                & ( N2 = N2 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N2 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N2 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_1317_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N2: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_nat = none_nat )
                & ( N2 = N2 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N2 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N2 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_1318_div__exp__eq,axiom,
    ! [A: uint32,M: nat,N2: nat] :
      ( ( divide_divide_uint32 @ ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_1319_div__exp__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,N2: nat] :
      ( ( divide1791077408188789448l_num1 @ ( divide1791077408188789448l_num1 @ A @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide1791077408188789448l_num1 @ A @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_1320_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_1321_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_1322_semiring__norm_I90_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(90)
thf(fact_1323_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_1324_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_1325_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_1326_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_1327_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_1328_bits__div__by__1,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ A @ one_on7727431528512463931l_num1 )
      = A ) ).

% bits_div_by_1
thf(fact_1329_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_1330_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_1331_semiring__norm_I88_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N2 ) ) ).

% semiring_norm(88)
thf(fact_1332_semiring__norm_I89_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N2 ) ) ).

% semiring_norm(89)
thf(fact_1333_semiring__norm_I84_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit1 @ N2 ) ) ).

% semiring_norm(84)
thf(fact_1334_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_1335_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_1336_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_1337_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri8819519690708144855l_num1 @ ( numeral_numeral_nat @ N2 ) )
      = ( numera7442385471795722001l_num1 @ N2 ) ) ).

% of_nat_numeral
thf(fact_1338_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% of_nat_numeral
thf(fact_1339_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% of_nat_numeral
thf(fact_1340_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% of_nat_numeral
thf(fact_1341_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% of_nat_numeral
thf(fact_1342_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% of_nat_numeral
thf(fact_1343_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_1344_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_1345_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_1346_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_1347_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_1348_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_1349_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_1350_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_1351_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_1352_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_1353_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_1354_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_1355_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_1356_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_1357_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_1358_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_1359_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_1360_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_1361_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_1362_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_1363_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_1364_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_1365_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_1366_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_1367_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_1368_of__nat__1,axiom,
    ( ( semiri8819519690708144855l_num1 @ one_one_nat )
    = one_on7727431528512463931l_num1 ) ).

% of_nat_1
thf(fact_1369_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_1370_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_1371_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_1372_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_1373_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_1374_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_1375_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_1376_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_1377_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_1378_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_1379_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X2 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1380_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X2 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1381_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X2 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1382_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri4939895301339042750nteger @ X2 )
        = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1383_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1384_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1385_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1386_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W )
        = ( semiri4939895301339042750nteger @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1387_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri2565882477558803405uint32 @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_uint32 @ ( semiri2565882477558803405uint32 @ M ) @ N2 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_1388_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8819519690708144855l_num1 @ ( power_power_nat @ M @ N2 ) )
      = ( power_2184487114949457152l_num1 @ ( semiri8819519690708144855l_num1 @ M ) @ N2 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_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_1398_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_1399_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_1400_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_1401_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8819519690708144855l_num1 @ ( suc @ M ) )
      = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( semiri8819519690708144855l_num1 @ M ) ) ) ).

% of_nat_Suc
thf(fact_1402_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_1403_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_1404_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_1405_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_1406_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_1407_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_1408_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_1409_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_1410_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_1411_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_1412_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_1413_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_1414_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_1415_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_1416_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_1417_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_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_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_1425_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_1426_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_1427_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_1428_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1429_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1430_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1431_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1432_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1433_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1434_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1435_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1436_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1437_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1438_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1439_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1440_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1441_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1442_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1443_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1444_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1445_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1446_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1447_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1448_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_1449_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_1450_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_1451_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_1452_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_1453_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_1454_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_1455_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_1456_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_1457_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_1458_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_1459_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_1460_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_1461_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_1462_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_1463_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_1464_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_1465_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_1466_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_1467_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_1468_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_1469_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_1470_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_1471_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_1472_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_1473_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_1474_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_1475_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_1476_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) )
     => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ).

% hoare_triple_preI
thf(fact_1477_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) )
     => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ).

% hoare_triple_preI
thf(fact_1478_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => ( hoare_hoare_triple_o @ P @ C @ Q ) )
     => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ).

% hoare_triple_preI
thf(fact_1479_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ).

% hoare_triple_preI
thf(fact_1480_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => ( hoare_8945653483474564448t_unit @ P @ C @ Q ) )
     => ( hoare_8945653483474564448t_unit @ P @ C @ Q ) ) ).

% hoare_triple_preI
thf(fact_1481_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 ) ) )
         => ( ! [M2: num] :
                ( X2
               != ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
           => ( ! [M2: num,N4: num] :
                  ( X2
                 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N4 ) ) )
             => ( ! [M2: num,N4: num] :
                    ( X2
                   != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N4 ) ) )
               => ( ! [M2: num] :
                      ( X2
                     != ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
                 => ( ! [M2: num,N4: num] :
                        ( X2
                       != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N4 ) ) )
                   => ~ ! [M2: num,N4: num] :
                          ( X2
                         != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_1482_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_1483_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_1484_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_1485_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_1486_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_1487_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_1488_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_1489_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_1490_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_1491_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_1492_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_1493_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_1494_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_1495_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_1496_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_1497_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_1498_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_1499_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_1500_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_1501_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_1502_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_1503_num_Oexhaust,axiom,
    ! [Y2: num] :
      ( ( Y2 != one )
     => ( ! [X24: num] :
            ( Y2
           != ( bit0 @ X24 ) )
       => ~ ! [X32: num] :
              ( Y2
             != ( bit1 @ X32 ) ) ) ) ).

% num.exhaust
thf(fact_1504_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_1505_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N: nat,M7: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M7 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_1506_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat,M7: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M7 ) ) ) ) ).

% nat_less_real_le
thf(fact_1507_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_1508_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_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_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_1518_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X6: real] : ( member_real @ X6 @ S2 )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S2 )
           => ( ord_less_eq_real @ X3 @ Z5 ) )
       => ? [Y3: real] :
            ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ord_less_eq_real @ X6 @ Y3 ) )
            & ! [Z5: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S2 )
                   => ( ord_less_eq_real @ X3 @ Z5 ) )
               => ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_1519_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_1520_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_1521_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_1522_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_1523_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_1524_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_1525_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_1526_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_1527_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_1528_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_1529_power__numeral__odd,axiom,
    ! [Z: uint32,W: num] :
      ( ( power_power_uint32 @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_uint32 @ ( times_times_uint32 @ Z @ ( power_power_uint32 @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_uint32 @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1530_power__numeral__odd,axiom,
    ! [Z: word_N3645301735248828278l_num1,W: num] :
      ( ( power_2184487114949457152l_num1 @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ Z @ ( power_2184487114949457152l_num1 @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_2184487114949457152l_num1 @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_power3__eq__cube,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_uint32 @ ( times_times_uint32 @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1539_power3__eq__cube,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1540_power3__eq__cube,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1541_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1542_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1543_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1544_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_1545_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_1546_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_1547_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_1548_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_1549_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_1550_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_1551_mult__diff__mult,axiom,
    ! [X2: real,Y2: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X2 @ Y2 ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y2 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1552_mult__diff__mult,axiom,
    ! [X2: rat,Y2: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X2 @ Y2 ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X2 @ ( minus_minus_rat @ Y2 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X2 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1553_mult__diff__mult,axiom,
    ! [X2: int,Y2: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X2 @ Y2 ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X2 @ ( minus_minus_int @ Y2 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X2 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1554_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_1555_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_1556_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_1557_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_1558_space__bound,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space_bound
thf(fact_1559_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( vEBT_VEBT_cnt2 @ T2 ) ) ) ) ).

% cnt_cnt_eq
thf(fact_1560_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( 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_1561_space__cnt,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_cnt @ T ) ) ) ).

% space_cnt
thf(fact_1562_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space'_bound
thf(fact_1563_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_1564_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_1565_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_1566_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_1567_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% delete_bound_height
thf(fact_1568_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_1569_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_1570_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_1571_TBOUND__highi,axiom,
    ! [X2: nat,N2: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_highi @ X2 @ N2 ) @ one_one_nat ) ).

% TBOUND_highi
thf(fact_1572_TBOUND__lowi,axiom,
    ! [X2: nat,N2: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_lowi @ X2 @ N2 ) @ one_one_nat ) ).

% TBOUND_lowi
thf(fact_1573_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_maxti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_maxti
thf(fact_1574_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_minti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_minti
thf(fact_1575_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_1576_space__space_H,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).

% space_space'
thf(fact_1577_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_1578_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_succi
thf(fact_1579_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_1580_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_1581_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_1582_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_1583_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_1584_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_1585_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : ( time_TBOUND_o @ ( vEBT_VEBT_minNulli @ T ) @ one_one_nat ) ).

% TBOUND_minNulli
thf(fact_1586_mod__pure__star__dist,axiom,
    ! [P: assn,B: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & B ) ) ).

% mod_pure_star_dist
thf(fact_1587_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_1588_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_1589_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X2 ) @ ( if_nat @ ( vEBT_VEBT_minNull @ T ) @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ) ).

% TBOUND_vebt_inserti
thf(fact_1590_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1591_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_1592_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).

% not_min_Null_member
thf(fact_1593_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X2 ) ) ).

% min_Null_member
thf(fact_1594_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% deg_not_0
thf(fact_1595_T__vebt__buildupi__gq__0,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_1596_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_1597_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_1598_TBOUND__vebt__buildupi,axiom,
    ! [N2: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% TBOUND_vebt_buildupi
thf(fact_1599_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_1600_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_1601_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X2 ) @ one_one_nat ) ) ).

% TBOUND_minNull
thf(fact_1602_ex__assn__const,axiom,
    ! [C: assn] :
      ( ( ex_ass463751140784270563_VEBTi
        @ ^ [X: list_VEBT_VEBTi] : C )
      = C ) ).

% ex_assn_const
thf(fact_1603_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_1604_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_1605_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_1606_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1607_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1608_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1609_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1610_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1611_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1612_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1613_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1614_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_1615_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_1616_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_1617_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_1618_mult__zero__right,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ zero_zero_uint32 )
      = zero_zero_uint32 ) ).

% mult_zero_right
thf(fact_1619_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1620_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1621_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1622_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1623_mult__zero__left,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ zero_zero_uint32 @ A )
      = zero_zero_uint32 ) ).

% mult_zero_left
thf(fact_1624_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1625_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1626_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1627_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1628_bits__div__by__0,axiom,
    ! [A: uint32] :
      ( ( divide_divide_uint32 @ A @ zero_zero_uint32 )
      = zero_zero_uint32 ) ).

% bits_div_by_0
thf(fact_1629_bits__div__by__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ A @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_div_by_0
thf(fact_1630_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1631_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1632_bits__div__0,axiom,
    ! [A: uint32] :
      ( ( divide_divide_uint32 @ zero_zero_uint32 @ A )
      = zero_zero_uint32 ) ).

% bits_div_0
thf(fact_1633_bits__div__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ zero_z3563351764282998399l_num1 @ A )
      = zero_z3563351764282998399l_num1 ) ).

% bits_div_0
thf(fact_1634_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1635_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1636_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1637_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1638_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1639_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1640_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1641_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1642_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1643_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1644_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1645_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_1646_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1647_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_1648_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1649_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_1650_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1651_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1652_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1653_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_VEBT_minNull @ T ) ) ) ) ) ).

% minNulli_rule
thf(fact_1654_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_1655_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_1656_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1657_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_1658_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      ( ( times_times_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_star
thf(fact_1659_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_1660_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_1661_mod__ex__dist,axiom,
    ! [P: list_VEBT_VEBTi > assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ H2 )
      = ( ? [X: list_VEBT_VEBTi] : ( rep_assn @ ( P @ X ) @ H2 ) ) ) ).

% mod_ex_dist
thf(fact_1662_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1663_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1664_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1665_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1666_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1667_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1668_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_1669_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_1670_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_1671_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_1672_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1673_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_1674_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_1675_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_1676_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_1677_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_1678_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1679_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1680_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_1681_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1682_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_1683_diff__numeral__special_I9_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ one_on7727431528512463931l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% diff_numeral_special(9)
thf(fact_1684_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_uint32 @ one_one_uint32 @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% diff_numeral_special(9)
thf(fact_1685_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1686_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1687_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1688_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1689_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1690_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1691_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1692_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1693_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1694_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1695_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1696_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1697_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1698_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1699_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1700_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1701_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1702_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1703_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1704_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1705_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1706_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_1707_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_uint32 @ zero_zero_uint32 @ ( suc @ N2 ) )
      = zero_zero_uint32 ) ).

% power_0_Suc
thf(fact_1708_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( suc @ N2 ) )
      = zero_z3563351764282998399l_num1 ) ).

% power_0_Suc
thf(fact_1709_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1710_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1711_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1712_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_1713_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_uint32 @ zero_zero_uint32 @ ( numeral_numeral_nat @ K ) )
      = zero_zero_uint32 ) ).

% power_zero_numeral
thf(fact_1714_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( numeral_numeral_nat @ K ) )
      = zero_z3563351764282998399l_num1 ) ).

% power_zero_numeral
thf(fact_1715_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1716_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1717_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1718_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_1719_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_1720_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_1721_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_1722_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1723_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_1724_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_1725_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_1726_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_1727_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_1728_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri2565882477558803405uint32 @ zero_zero_nat )
    = zero_zero_uint32 ) ).

% semiring_1_class.of_nat_0
thf(fact_1729_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_1730_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_1731_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_1732_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_1733_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% semiring_1_class.of_nat_0
thf(fact_1734_power__Suc0__right,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1735_power__Suc0__right,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1736_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1737_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1738_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1739_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_1740_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1741_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_1742_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_1743_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_1744_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_1745_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_1746_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_1747_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1748_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_1749_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_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_1757_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_1758_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1759_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1760_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1761_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1762_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_1763_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_1764_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_1765_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_1766_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_1767_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_1768_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_1769_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_1770_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_1771_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_1772_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_1773_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_1774_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_1775_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( power_power_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1776_power__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( power_power_real @ A @ N2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1777_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1778_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ( power_power_int @ A @ N2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1779_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_1780_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_1781_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_1782_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_1783_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_1784_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_1785_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_1786_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_1787_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_1788_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_1789_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_1790_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1791_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1792_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1793_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1794_power__mono__iff,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1795_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1796_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1797_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1798_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_1799_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_1800_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_1801_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_1802_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_1803_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_1804_bits__1__div__2,axiom,
    ( ( divide_divide_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = zero_zero_uint32 ) ).

% bits_1_div_2
thf(fact_1805_bits__1__div__2,axiom,
    ( ( divide1791077408188789448l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
    = zero_z3563351764282998399l_num1 ) ).

% bits_1_div_2
thf(fact_1806_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_1807_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_1808_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_1809_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_1810_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_1811_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_1812_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_1813_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_1814_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_1815_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_1816_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_1817_zero__less__power2,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_1818_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_1819_zero__less__power2,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_1820_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_1821_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_1822_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_1823_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1824_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1825_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1826_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1827_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_1828_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_1829_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_1830_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_1831_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_1832_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X2 ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% htt_vebt_inserti
thf(fact_1833_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_1834_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_1835_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_1836_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_1837_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_1838_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_1839_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_1840_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_1841_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_1842_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_uint32 @ zero_zero_uint32 @ N2 )
        = zero_zero_uint32 ) ) ).

% zero_power
thf(fact_1843_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_1844_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_1845_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_1846_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_1847_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1848_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1849_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1850_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1851_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_1852_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_1853_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1854_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_1855_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1856_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1857_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1858_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1859_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1860_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1861_zero__neq__one,axiom,
    zero_z3563351764282998399l_num1 != one_on7727431528512463931l_num1 ).

% zero_neq_one
thf(fact_1862_zero__neq__one,axiom,
    zero_zero_uint32 != one_one_uint32 ).

% zero_neq_one
thf(fact_1863_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1864_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1865_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1866_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1867_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1868_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1869_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1870_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1871_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1872_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1873_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1874_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1875_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( times_times_real @ A @ B )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_1876_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_1877_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_1878_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_1879_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_1880_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_1881_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_1882_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_1883_mult__not__zero,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( times_times_uint32 @ A @ B )
       != zero_zero_uint32 )
     => ( ( A != zero_zero_uint32 )
        & ( B != zero_zero_uint32 ) ) ) ).

% mult_not_zero
thf(fact_1884_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_1885_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_1886_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_1887_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_1888_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N2 )
       != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_1889_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N2 )
       != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_1890_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,N2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N2 )
       != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_1891_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N2 )
       != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_1892_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1893_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M2: nat] :
          ( N2
          = ( suc @ M2 ) ) ) ).

% not0_implies_Suc
thf(fact_1894_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1895_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1896_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1897_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_1898_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_1899_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_1900_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y2
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1901_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat
        = ( suc @ X23 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1902_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1903_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1904_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( zero_zero_nat
     != ( suc @ X23 ) ) ).

% nat.distinct(1)
thf(fact_1905_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N4 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_1906_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1907_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1908_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_1909_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1910_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_1911_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1912_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1913_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1914_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1915_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1916_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_1917_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_1918_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1919_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_1920_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_1921_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_1922_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Leafi @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBTi.size(4)
thf(fact_1923_power__eq__imp__eq__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ N2 )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1924_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N2 )
        = ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1925_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1926_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ N2 )
        = ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1927_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N2 )
              = ( power_power_real @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1928_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N2 )
              = ( power_power_rat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1929_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N2 )
              = ( power_power_nat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1930_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N2 )
              = ( power_power_int @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1931_lambda__zero,axiom,
    ( ( ^ [H: uint32] : zero_zero_uint32 )
    = ( times_times_uint32 @ zero_zero_uint32 ) ) ).

% lambda_zero
thf(fact_1932_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_1933_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_1934_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_1935_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_1936_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_1937_VEBT__internal_OminNulli_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_minNulli @ ( vEBT_Leafi @ $false @ $false ) )
    = ( heap_Time_return_o @ $true ) ) ).

% VEBT_internal.minNulli.simps(1)
thf(fact_1938_VEBT__internal_OminNulli_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Leafi @ $true @ Uv ) )
      = ( heap_Time_return_o @ $false ) ) ).

% VEBT_internal.minNulli.simps(2)
thf(fact_1939_VEBT__internal_OminNulli_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Leafi @ Uu @ $true ) )
      = ( heap_Time_return_o @ $false ) ) ).

% VEBT_internal.minNulli.simps(3)
thf(fact_1940_power__strict__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_1941_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_1942_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_1943_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_1944_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_1945_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_1946_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_1947_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_1948_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_1949_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1950_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1951_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_1952_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc2 ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_1953_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_1954_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_1955_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_1956_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_1957_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_1958_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1959_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1960_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_1961_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1962_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1963_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1964_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1965_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_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1974_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1975_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1976_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1977_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1978_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1979_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1980_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1981_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1982_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1983_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1984_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1985_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1986_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1987_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1988_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1989_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1990_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1991_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1992_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1993_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1994_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1995_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1996_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_1997_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_1998_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_1999_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2000_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2001_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2002_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2003_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2004_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2005_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2006_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2007_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2008_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2009_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2010_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2011_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2012_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2013_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2014_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2015_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2016_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2017_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2018_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2019_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2020_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2021_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2022_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2023_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2024_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2025_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2026_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2027_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2028_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2029_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2030_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2031_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2032_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2033_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2034_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_2035_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_2036_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_2037_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2038_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2039_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2040_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2041_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2042_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2043_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2044_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2045_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2046_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2047_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2048_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2049_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2050_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2051_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2052_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2053_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2054_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2055_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2056_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2057_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2058_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2059_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2060_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2061_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2062_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2063_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2064_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2065_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2066_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2067_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2068_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2069_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2070_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2071_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2072_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2073_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2074_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2075_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2076_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2077_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2078_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2079_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2080_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2081_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2082_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2083_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2084_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2085_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2086_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2087_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2088_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2089_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2090_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2091_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2092_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2093_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2094_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2095_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2096_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2097_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2098_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2099_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2100_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2101_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2102_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2103_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2104_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2105_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2106_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2107_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2108_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2109_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2110_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2111_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2112_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2113_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2114_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2115_zero__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2116_zero__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2117_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2118_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2119_power__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2120_power__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2121_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2122_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2123_zero__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2124_zero__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2125_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2126_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2127_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_2128_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_2129_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_2130_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_2131_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_2132_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_2133_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_2134_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_2135_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_2136_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_2137_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_2138_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_2139_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_2140_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_2141_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_2142_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_2143_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2144_power__0,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ A @ zero_zero_nat )
      = one_one_uint32 ) ).

% power_0
thf(fact_2145_power__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ A @ zero_zero_nat )
      = one_on7727431528512463931l_num1 ) ).

% power_0
thf(fact_2146_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2147_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2148_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2149_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_2150_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M2: nat] :
          ( N2
          = ( suc @ M2 ) ) ) ).

% gr0_implies_Suc
thf(fact_2151_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_2152_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M7: nat] :
            ( N2
            = ( suc @ M7 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2153_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_2154_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_2155_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_2156_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_2157_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_2158_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2159_option_Osize_I4_J,axiom,
    ! [X23: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X23 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2160_option_Osize_I4_J,axiom,
    ! [X23: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X23 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2161_option_Osize_I4_J,axiom,
    ! [X23: num] :
      ( ( size_size_option_num @ ( some_num @ X23 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2162_option_Osize_I3_J,axiom,
    ( ( size_s3991424295186984831it_nat @ none_P1551326421579882414it_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2163_option_Osize_I3_J,axiom,
    ( ( size_s364044314319911927it_nat @ none_P7668321371905463026it_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2164_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2165_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2166_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2167_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2168_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2169_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_2170_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_2171_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_2172_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_2173_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_2174_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_2175_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_2176_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_2177_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_2178_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_2179_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_2180_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_2181_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ X2 )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) ) ).

% vebt_insert.simps(2)
thf(fact_2182_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_2183_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_2184_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_2185_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2186_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2187_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2188_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2189_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2190_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2191_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2192_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2193_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2194_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2195_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2196_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2197_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2198_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2199_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2200_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2201_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2202_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2203_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2204_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2205_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2206_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2207_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2208_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2209_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2210_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2211_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2212_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2213_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2214_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2215_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2216_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2217_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2218_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2219_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2220_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2221_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2222_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2223_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2224_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2225_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2226_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2227_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2228_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2229_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2230_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2231_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2232_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2233_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2234_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2235_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_2236_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_2237_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_2238_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_2239_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_2240_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_2241_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2242_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2243_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2244_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2245_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_2246_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_2247_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_2248_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_2249_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_2250_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_2251_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_2252_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_2253_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_2254_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_2255_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2256_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2257_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2258_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2259_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2260_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2261_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2262_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2263_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_2264_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_2265_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_2266_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_2267_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_2268_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_2269_power__less__imp__less__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2270_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2271_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2272_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2273_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2274_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2275_power__le__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2276_power__le__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2277_power__le__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2278_power__le__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2279_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2280_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2281_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2282_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2283_power__le__imp__le__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2284_power__le__imp__le__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2285_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2286_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2287_power__inject__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N2 ) )
        = ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2288_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
        = ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2289_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
        = ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2290_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N2 ) )
        = ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2291_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2292_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2293_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2294_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2295_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc2 ) )
      = ( heap_Time_return_o @ $false ) ) ).

% VEBT_internal.minNulli.simps(5)
thf(fact_2296_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_2297_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
      = ( heap_Time_return_o @ $true ) ) ).

% VEBT_internal.minNulli.simps(4)
thf(fact_2298_num_Osize_I5_J,axiom,
    ! [X23: num] :
      ( ( size_size_num @ ( bit0 @ X23 ) )
      = ( plus_plus_nat @ ( size_size_num @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_2299_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_2300_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_2301_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_2302_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_2303_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_2304_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_2305_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_2306_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_2307_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2308_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_2309_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_2310_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_2311_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_2312_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_2313_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_2314_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_2315_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_2316_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A @ C ) @ B ) ) ) ).

% td_gal_lt
thf(fact_2317_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P @ Q ) @ R2 )
      = ( times_times_assn @ P @ ( times_times_assn @ Q @ R2 ) ) ) ).

% assn_times_assoc
thf(fact_2318_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P3: assn,Q4: assn] : ( times_times_assn @ Q4 @ P3 ) ) ) ).

% assn_times_comm
thf(fact_2319_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X2 )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) ).

% vebt_insert.simps(3)
thf(fact_2320_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 ) ).

% vebt_member.simps(3)
thf(fact_2321_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_2322_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_2323_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,Vc2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_2324_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2325_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2326_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2327_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2328_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2329_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2330_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2331_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2332_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2333_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2334_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2335_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2336_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2337_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2338_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2339_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2340_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2341_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2342_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2343_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2344_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2345_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2346_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2347_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2348_convex__bound__le,axiom,
    ! [X2: real,A: real,Y2: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X2 @ A )
     => ( ( ord_less_eq_real @ Y2 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2349_convex__bound__le,axiom,
    ! [X2: rat,A: rat,Y2: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X2 @ A )
     => ( ( ord_less_eq_rat @ Y2 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2350_convex__bound__le,axiom,
    ! [X2: int,A: int,Y2: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X2 @ A )
     => ( ( ord_less_eq_int @ Y2 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2351_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2352_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2353_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2354_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2355_power__Suc__less,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2356_power__Suc__less,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2357_power__Suc__less,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2358_power__Suc__less,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2359_power__Suc__le__self,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2360_power__Suc__le__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2361_power__Suc__le__self,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2362_power__Suc__le__self,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2363_power__Suc__less__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2364_power__Suc__less__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2365_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2366_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2367_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2368_zero__power2,axiom,
    ( ( power_power_uint32 @ zero_zero_uint32 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_uint32 ) ).

% zero_power2
thf(fact_2369_zero__power2,axiom,
    ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3563351764282998399l_num1 ) ).

% zero_power2
thf(fact_2370_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2371_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2372_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2373_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2374_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2375_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2376_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2377_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2378_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2379_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2380_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2381_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_2382_self__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2383_self__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2384_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2385_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2386_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2387_one__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2388_one__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2389_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2390_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2391_power__diff,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2392_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2393_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2394_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2395_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_2396_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_2397_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_2398_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_2399_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M7: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M7 @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M7 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_2400_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_2401_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M7: nat,N: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M7 @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_2402_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_2403_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_2404_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_2405_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_2406_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 )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
               => ( P @ I2 ) ) ) ) ) ) ).

% split_div
thf(fact_2407_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B @ C ) @ A )
        = ( ord_less_eq_nat @ B @ ( divide_divide_nat @ A @ C ) ) ) ) ).

% td_gal
thf(fact_2408_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M7: nat,N: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M7 @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_2409_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ B @ N2 ) ) @ ( power_power_nat @ B @ M ) )
        = ( ord_less_nat @ A @ ( power_power_nat @ B @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_2410_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ X2 ) ).

% vebt_member.simps(4)
thf(fact_2411_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_2412_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_2413_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_2414_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,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_2415_convex__bound__lt,axiom,
    ! [X2: real,A: real,Y2: real,U: real,V: real] :
      ( ( ord_less_real @ X2 @ A )
     => ( ( ord_less_real @ Y2 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2416_convex__bound__lt,axiom,
    ! [X2: rat,A: rat,Y2: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X2 @ A )
     => ( ( ord_less_rat @ Y2 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2417_convex__bound__lt,axiom,
    ! [X2: int,A: int,Y2: int,U: int,V: int] :
      ( ( ord_less_int @ X2 @ A )
     => ( ( ord_less_int @ Y2 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2418_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_2419_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_2420_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_2421_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_2422_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% half_gt_zero_iff
thf(fact_2423_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_2424_half__gt__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2425_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2426_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_2427_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_2428_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_2429_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_2430_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_2431_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_2432_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_2433_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_2434_zero__le__power2,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2435_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2436_zero__le__power2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2437_power2__less__0,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_2438_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_2439_power2__less__0,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_2440_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_uint32 )
     => ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_uint32 ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2441_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_2442_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_2443_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_2444_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_uint32 )
     => ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M )
       != zero_zero_uint32 ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2445_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_2446_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_2447_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_2448_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_uint32 )
     => ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_uint32 ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2449_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_2450_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_2451_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_2452_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2453_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2454_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_2455_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_2456_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_2457_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_2458_power__eq__if,axiom,
    ( power_power_uint32
    = ( ^ [P5: uint32,M7: nat] : ( if_uint32 @ ( M7 = zero_zero_nat ) @ one_one_uint32 @ ( times_times_uint32 @ P5 @ ( power_power_uint32 @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2459_power__eq__if,axiom,
    ( power_2184487114949457152l_num1
    = ( ^ [P5: word_N3645301735248828278l_num1,M7: nat] : ( if_wor5778924947035936048l_num1 @ ( M7 = zero_zero_nat ) @ one_on7727431528512463931l_num1 @ ( times_7065122842183080059l_num1 @ P5 @ ( power_2184487114949457152l_num1 @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2460_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M7: nat] : ( if_real @ ( M7 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2461_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M7: nat] : ( if_rat @ ( M7 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2462_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M7: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2463_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M7: nat] : ( if_int @ ( M7 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2464_power__minus__mult,axiom,
    ! [N2: nat,A: uint32] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_uint32 @ ( power_power_uint32 @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_uint32 @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2465_power__minus__mult,axiom,
    ! [N2: nat,A: word_N3645301735248828278l_num1] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_2184487114949457152l_num1 @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2466_power__minus__mult,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2467_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2468_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2469_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2470_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_2471_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 ) )
        | ? [Q5: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q5 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q5 ) ) )
            & ( P @ Q5 ) ) ) ) ).

% split_div'
thf(fact_2472_power__sub,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_sub
thf(fact_2473_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_2474_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_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_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_2486_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_2487_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_2488_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_2489_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_2490_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_2491_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_2492_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_2493_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 ) ) )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leafi @ Uu2 @ $true ) )
             => ( Y2
               != ( heap_Time_return_o @ $false ) ) )
           => ( ( ? [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ( Y2
                 != ( heap_Time_return_o @ $true ) ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( X2
                      = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => ( Y2
                   != ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
thf(fact_2494_zero__le__even__power_H,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2495_zero__le__even__power_H,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2496_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2497_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_2498_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_2499_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_2500_vebt__minti_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_minti.simps(1)
thf(fact_2501_vebt__maxti_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_maxti.simps(1)
thf(fact_2502_mod__starE,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
     => ~ ( ? [X_12: produc3658429121746597890et_nat] : ( rep_assn @ A @ X_12 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B @ H_2 ) ) ) ).

% mod_starE
thf(fact_2503_mod__starD,axiom,
    ! [A2: assn,B5: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A2 @ B5 ) @ H2 )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A2 @ H1 )
          & ( rep_assn @ B5 @ H22 ) ) ) ).

% mod_starD
thf(fact_2504_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2505_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2506_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2507_odd__power__less__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_2508_odd__power__less__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_2509_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_2510_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_2511_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_2512_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_2513_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_2514_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_2515_ex__distrib__star,axiom,
    ! [P: list_VEBT_VEBTi > assn,Q: assn] :
      ( ( ex_ass463751140784270563_VEBTi
        @ ^ [X: list_VEBT_VEBTi] : ( times_times_assn @ ( P @ X ) @ Q ) )
      = ( times_times_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ Q ) ) ).

% ex_distrib_star
thf(fact_2516_ex__one__point__gen,axiom,
    ! [P: list_VEBT_VEBTi > assn,V: list_VEBT_VEBTi] :
      ( ! [H3: produc3658429121746597890et_nat,X3: list_VEBT_VEBTi] :
          ( ( rep_assn @ ( P @ X3 ) @ H3 )
         => ( X3 = V ) )
     => ( ( ex_ass463751140784270563_VEBTi @ P )
        = ( P @ V ) ) ) ).

% ex_one_point_gen
thf(fact_2517_mod__exI,axiom,
    ! [P: list_VEBT_VEBTi > assn,H2: produc3658429121746597890et_nat] :
      ( ? [X6: list_VEBT_VEBTi] : ( rep_assn @ ( P @ X6 ) @ H2 )
     => ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ H2 ) ) ).

% mod_exI
thf(fact_2518_mod__exE,axiom,
    ! [P: list_VEBT_VEBTi > assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ H2 )
     => ~ ! [X3: list_VEBT_VEBTi] :
            ~ ( rep_assn @ ( P @ X3 ) @ H2 ) ) ).

% mod_exE
thf(fact_2519_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_2520_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_2521_vebt__minti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X2 )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leafi @ A3 @ B2 ) )
           => ~ ( ( A3
                 => ( Y2
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                & ( ~ A3
                 => ( ( B2
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                    & ( ~ B2
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X2
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) ) ) ) ) ) ).

% vebt_minti.elims
thf(fact_2522_vebt__maxti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X2 )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leafi @ A3 @ B2 ) )
           => ~ ( ( B2
                 => ( Y2
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                & ( ~ B2
                 => ( ( A3
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                    & ( ~ A3
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X2
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) ) ) ) ) ) ).

% vebt_maxti.elims
thf(fact_2523_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_2524_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_VEBT_minNull @ T ) ) ) )
      @ one_one_nat ) ).

% minNrulli_ruleT
thf(fact_2525_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_2526_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2527_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2528_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2529_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2530_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2531_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2532_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2533_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2534_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_2535_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2536_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2537_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2538_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2539_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_2540_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_2541_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).

% cnt_non_neg
thf(fact_2542_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T2: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).

% Tb_Tb'
thf(fact_2543_time__replicate,axiom,
    ! [X2: heap_T5738788834812785303t_unit,C: nat,N2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ X2 @ H3 ) @ 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_2544_time__replicate,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,C: nat,N2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ X2 @ H3 ) @ 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_2545_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_2546_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_2547_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2548_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2549_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2550_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2551_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_2552_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_2553_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2554_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2555_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2556_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2557_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2558_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2559_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2560_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2561_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_2562_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_2563_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_2564_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = A )
        = ( B = one_one_int ) ) ) ).

% int_div_same_is_1
thf(fact_2565_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2566_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2567_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2568_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2569_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_2570_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_2571_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_2572_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_2573_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_2574_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_2575_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_2576_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_2577_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_2578_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_2579_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_2580_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_2581_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2582_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2583_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2584_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2585_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2586_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2587_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2588_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2589_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2590_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2591_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_2592_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_2593_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_2594_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_2595_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_2596_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_2597_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2598_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2599_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2600_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2601_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2602_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2603_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2604_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2605_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_2606_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_2607_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
      @ ^ [R: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_member @ T @ X2 ) ) ) )
      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% htt_vebt_memberi
thf(fact_2608_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_2609_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_2610_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_2611_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_2612_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_2613_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_2614_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_2615_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_2616_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_2617_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_2618_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_2619_zdiv__mono2__neg,axiom,
    ! [A: int,B3: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_2620_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( ( ord_less_eq_int @ A @ A5 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_2621_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_2622_zdiv__mono2,axiom,
    ! [A: int,B3: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_2623_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( ( ord_less_eq_int @ A @ A5 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_2624_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ A ) ) ) ).

% zdiv_le_dividend
thf(fact_2625_q__pos__lemma,axiom,
    ! [B3: int,Q6: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B3 )
       => ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).

% q_pos_lemma
thf(fact_2626_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R3: int,B3: int,Q6: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q2 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_2627_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R3: int,B3: int,Q6: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q6 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q6 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_2628_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R3 @ B )
           => ( ord_less_eq_int @ Q6 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_2629_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q2 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_2630_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_2631_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_2632_linordered__field__no__lb,axiom,
    ! [X6: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X6 ) ).

% linordered_field_no_lb
thf(fact_2633_linordered__field__no__lb,axiom,
    ! [X6: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X6 ) ).

% linordered_field_no_lb
thf(fact_2634_linordered__field__no__ub,axiom,
    ! [X6: real] :
    ? [X_12: real] : ( ord_less_real @ X6 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_2635_linordered__field__no__ub,axiom,
    ! [X6: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X6 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_2636_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_2637_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R5: real] :
          ( ( ord_less_real @ zero_zero_real @ R5 )
          & ( ( power_power_real @ R5 @ ( suc @ N2 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_2638_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_2639_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_2640_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_2641_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 )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) ) ) ) ).

% split_zdiv
thf(fact_2642_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_2643_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_2644_zdiv__mult__self,axiom,
    ! [M: int,A: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ M @ N2 ) ) @ M )
        = ( plus_plus_int @ ( divide_divide_int @ A @ M ) @ N2 ) ) ) ).

% zdiv_mult_self
thf(fact_2645_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N2 )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N2 )
                    = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_2646_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R5: real] :
            ( ( ord_less_real @ zero_zero_real @ R5 )
            & ( ( power_power_real @ R5 @ N2 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_2647_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_2648_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_2649_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 )
       => ( ! [M2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M2 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X2 ) @ C ) )
         => ( X2 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_2650_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_2651_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_2652_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_2653_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_2654_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_2655_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_2656_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_2657_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_2658_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2659_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2660_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_2661_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_2662_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_2663_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_2664_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_2665_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2666_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2667_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2668_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2669_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2670_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2671_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_2672_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_2673_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_2674_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_2675_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_2676_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_2677_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_2678_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_2679_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2680_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2681_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_2682_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_2683_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_2684_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_2685_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_2686_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_2687_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_2688_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_2689_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2690_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2691_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2692_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2693_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2694_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2695_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2696_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2697_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2698_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2699_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2700_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2701_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_2702_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_2703_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2704_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2705_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2706_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2707_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2708_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2709_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2710_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2711_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2712_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2713_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2714_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2715_field__le__epsilon,axiom,
    ! [X2: real,Y2: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y2 @ E2 ) ) )
     => ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% field_le_epsilon
thf(fact_2716_field__le__epsilon,axiom,
    ! [X2: rat,Y2: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y2 @ E2 ) ) )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% field_le_epsilon
thf(fact_2717_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_2718_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_2719_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_2720_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_2721_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_2722_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_2723_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2724_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2725_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_2726_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_2727_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_2728_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_2729_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_2730_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_2731_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_2732_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_2733_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2734_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2735_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2736_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2737_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2738_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2739_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2740_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2741_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2742_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2743_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2744_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2745_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2746_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2747_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2748_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2749_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_2750_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_2751_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_2752_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_2753_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2754_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2755_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2756_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2757_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2758_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2759_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2760_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2761_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_2762_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_2763_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_2764_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_2765_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_2766_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_2767_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_2768_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_2769_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_2770_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_2771_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).

% gt_half_sum
thf(fact_2772_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_2773_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_2774_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_2775_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2776_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2777_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_2778_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_2779_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_2780_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_2781_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_2782_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_2783_field__le__mult__one__interval,axiom,
    ! [X2: real,Y2: real] :
      ( ! [Z3: real] :
          ( ( ord_less_real @ zero_zero_real @ Z3 )
         => ( ( ord_less_real @ Z3 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X2 ) @ Y2 ) ) )
     => ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% field_le_mult_one_interval
thf(fact_2784_field__le__mult__one__interval,axiom,
    ! [X2: rat,Y2: rat] :
      ( ! [Z3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z3 )
         => ( ( ord_less_rat @ Z3 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X2 ) @ Y2 ) ) )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% field_le_mult_one_interval
thf(fact_2785_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2786_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2787_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2788_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2789_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2790_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2791_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2792_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2793_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2794_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2795_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2796_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2797_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2798_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2799_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2800_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2801_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2802_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2803_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_2804_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_2805_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_2806_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_2807_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2808_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2809_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_2810_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_2811_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_2812_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_2813_scaling__mono,axiom,
    ! [U: real,V: real,R3: real,S3: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R3 )
       => ( ( ord_less_eq_real @ R3 @ S3 )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R3 @ ( minus_minus_real @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2814_scaling__mono,axiom,
    ! [U: rat,V: rat,R3: rat,S3: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
       => ( ( ord_less_eq_rat @ R3 @ S3 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R3 @ ( minus_minus_rat @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2815_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_2816_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                  @ zero_zero_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_2817_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_2818_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_2819_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_2820_vebt__succ_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa )
        = Y2 )
     => ( ! [Uu2: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ Uu2 @ B2 ) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( B2
                   => ( Y2
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B2
                   => ( Y2 = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y2 != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2 != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( 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 @ TreeList2 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_2821_vebt__pred_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y2
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y2 = none_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ~ ( ( B2
                       => ( Y2
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B2
                       => ( ( A3
                           => ( Y2
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y2 = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( 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 @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_2822_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,TreeList: list_VEBT_VEBT,Summary: 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 ) @ TreeList @ Summary ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_2823_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,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X2 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_2824_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_2825_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( N2 = one_one_nat )
       => ? [A3: $o,B2: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).

% deg_1_Leafy
thf(fact_2826_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A3: $o,B2: $o] :
          ( T
          = ( vEBT_Leaf @ A3 @ B2 ) ) ) ).

% deg_1_Leaf
thf(fact_2827_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A4: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).

% deg1Leaf
thf(fact_2828_Tbuildupi__buildupi_H,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N2 ) )
      = ( vEBT_V9176841429113362141ildupi @ N2 ) ) ).

% Tbuildupi_buildupi'
thf(fact_2829_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_2830_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_2831_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_2832_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A @ B ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_2833_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_2834_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_2835_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X22 ) ) ).

% VEBT.distinct(1)
thf(fact_2836_VEBT_Oexhaust,axiom,
    ! [Y2: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y2
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X222: $o] :
            ( Y2
           != ( vEBT_Leaf @ X212 @ X222 ) ) ) ).

% VEBT.exhaust
thf(fact_2837_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_2838_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_2839_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_2840_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,
    ! [A: $o,B: $o,N2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( 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_2841_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,
    ! [A: $o,B: $o,N2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N2 ) ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_2842_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,
    ! [A: $o,B: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ 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_2843_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,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_2844_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,
    ! [A: $o,B: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( 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_2845_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,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_2846_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_2847_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_2848_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_2849_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 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_2850_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_2851_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_2852_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_2853_vebt__insert_Osimps_I1_J,axiom,
    ! [X2: nat,A: $o,B: $o] :
      ( ( ( X2 = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X2 != zero_zero_nat )
       => ( ( ( X2 = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X2 != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_2854_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_2855_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_2856_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( X2
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_2857_vebt__assn__raw_Ocases,axiom,
    ! [X2: produc3625547720036274456_VEBTi] :
      ( ! [A3: $o,B2: $o,Ai: $o,Bi: $o] :
          ( X2
         != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B2 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
     => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
            ( X2
           != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) )
       => ( ! [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_2858_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_2859_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_2860_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_2861_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_2862_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Ai2: $o,Bi2: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ A @ B ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) )
      = ( pure_assn
        @ ( ( Ai2 = A )
          & ( Bi2 = B ) ) ) ) ).

% vebt_assn_raw.simps(1)
thf(fact_2863_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_2864_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_2865_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_2866_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_2867_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_2868_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_2869_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_2870_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A3: $o,B2: $o] :
          ( X2
         != ( vEBT_Leaf @ A3 @ B2 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_2871_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_2872_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y2 )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y2 )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y2 )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y2 )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => Y2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_2873_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] :
      ( ! [A3: $o,B2: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,B2: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B2: $o,N4: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) @ X3 ) )
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_2874_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_2875_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_2876_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_2877_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_2878_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] :
      ( ! [A3: $o,B2: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X3 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_2879_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] :
      ( ! [A3: $o,B2: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X3 ) )
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_2880_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B2: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B2 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
         => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_2881_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw2: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B2: $o,Va3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_2882_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: 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 @ Va2 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ X3 ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_2883_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_2884_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ 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_2885_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_2886_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_2887_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_2888_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_2889_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_2890_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_2891_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_2892_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_2893_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_2894_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_2895_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_2896_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_2897_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_2898_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_2899_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N4: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% int_diff_cases
thf(fact_2900_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,Vc2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_2901_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% member_bound_height'
thf(fact_2902_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_2903_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_2904_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z2: int] :
        ? [N: nat] :
          ( Z2
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_2905_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 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y2 != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2 != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
               => ( Y2 != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( 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 @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_2906_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_2907_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_2908_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_2909_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_2910_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_2911_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_2912_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z2: int] :
        ? [N: nat] :
          ( Z2
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_2913_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_2914_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_2915_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_2916_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,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_2917_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,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_2918_vebt__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A3 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B2 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
             => ~ ( ( 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 @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_2919_vebt__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y2
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y2 )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y2 )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
               => Y2 )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( 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 @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_2920_vebt__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A3 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B2 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( 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 @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_2921_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 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( 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 ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
               => ( Y2
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( 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 @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_2922_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A4: $o,B4: $o] :
                ( A1
                = ( vEBT_Leaf @ A4 @ B4 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_2923_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A3: $o,B2: $o] :
              ( A12
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X6 @ N4 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                     => ( ( M2 = N4 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N4 @ M2 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X6: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X6 @ N4 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                       => ( ( M2
                            = ( suc @ N4 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N4 @ M2 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X6: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X6 @ N4 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                         => ( ( M2 = N4 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N4 @ M2 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X5 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X6: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                & ! [X6: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X6 @ N4 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N4 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X6 )
                                                      & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X6: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X6 @ N4 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                           => ( ( M2
                                = ( suc @ N4 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N4 @ M2 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X5 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X6: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                  & ! [X6: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X6 @ N4 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N4 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X6 )
                                                        & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_2924_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,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X2 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_2925_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,TreeList: list_VEBT_VEBT,Summary: 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 @ TreeList @ Summary ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_2926_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 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ A3 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ? [Va3: nat] :
                    ( Xa
                    = ( suc @ ( suc @ Va3 ) ) )
               => ( Y2 != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( 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 @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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_2927_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 )
     => ( ( ? [Uu2: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ B2 ) )
         => ( ( 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 ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( 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 @ TreeList2 ) )
                                @ ( if_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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_2928_vebt__maxt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( B2
                 => ( Y2
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B2
                 => ( ( A3
                     => ( Y2
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y2 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_2929_vebt__mint_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( A3
                 => ( Y2
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B2
                     => ( Y2
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B2
                     => ( Y2 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_2930_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 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y2 != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
           => ( Y2 != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y2 != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( 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 @ TreeList2 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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_2931_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_2932_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_2933_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2934_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2935_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2936_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2937_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2938_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2939_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2940_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2941_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2942_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2943_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2944_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2945_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2946_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2947_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2948_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2949_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2950_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_2951_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_2952_add__0,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% add_0
thf(fact_2953_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_2954_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_2955_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_2956_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_2957_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_2958_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_2959_add__cancel__right__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A
        = ( plus_plus_uint32 @ A @ B ) )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_right
thf(fact_2960_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_2961_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_2962_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_2963_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_2964_add__cancel__right__left,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A
        = ( plus_plus_uint32 @ B @ A ) )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_left
thf(fact_2965_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_2966_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_2967_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_2968_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_2969_add__cancel__left__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( plus_plus_uint32 @ A @ B )
        = A )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_right
thf(fact_2970_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_2971_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_2972_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_2973_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_2974_add__cancel__left__left,axiom,
    ! [B: uint32,A: uint32] :
      ( ( ( plus_plus_uint32 @ B @ A )
        = A )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_left
thf(fact_2975_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_2976_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_2977_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_2978_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_2979_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_2980_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2981_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2982_add_Oright__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% add.right_neutral
thf(fact_2983_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_2984_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_2985_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_2986_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_2987_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2988_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2989_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2990_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2991_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2992_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2993_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2994_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2995_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2996_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2997_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2998_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2999_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_3000_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_3001_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_3002_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_3003_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ A )
      = zero_zero_uint32 ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3004_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3005_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3006_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3007_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3008_diff__zero,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% diff_zero
thf(fact_3009_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_3010_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_3011_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_3012_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_3013_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3014_diff__0__right,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% diff_0_right
thf(fact_3015_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_3016_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_3017_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_3018_diff__self,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ A )
      = zero_zero_uint32 ) ).

% diff_self
thf(fact_3019_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_3020_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_3021_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_3022_mult_Oright__neutral,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A @ one_on7727431528512463931l_num1 )
      = A ) ).

% mult.right_neutral
thf(fact_3023_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_3024_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_3025_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_3026_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_3027_mult__1,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ one_on7727431528512463931l_num1 @ A )
      = A ) ).

% mult_1
thf(fact_3028_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_3029_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_3030_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_3031_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_3032_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3033_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3034_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3035_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3036_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3037_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3038_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3039_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3040_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3041_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3042_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3043_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3044_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3045_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3046_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3047_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3048_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3049_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3050_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3051_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3052_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3053_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3054_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_3055_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_3056_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_3057_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_3058_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_3059_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_3060_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_3061_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_3062_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_3063_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_3064_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_3065_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_3066_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_3067_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_3068_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_3069_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_3070_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3071_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3072_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3073_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3074_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3075_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3076_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_3077_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_3078_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_3079_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_3080_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_3081_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_3082_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_3083_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_3084_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_3085_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_3086_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_3087_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_3088_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_3089_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_3090_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_3091_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_3092_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3093_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3094_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3095_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3096_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3097_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3098_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_3099_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3100_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3101_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3102_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_3103_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3104_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3105_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3106_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3107_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3108_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3109_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3110_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3111_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_3112_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_3113_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_3114_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_3115_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_3116_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_3117_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_3118_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_3119_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_3120_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_3121_group__cancel_Oadd2,axiom,
    ! [B5: real,K: real,B: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B5 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3122_group__cancel_Oadd2,axiom,
    ! [B5: rat,K: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B5 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3123_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3124_group__cancel_Oadd2,axiom,
    ! [B5: int,K: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B5 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3125_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3126_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3127_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3128_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3129_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3130_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3131_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3132_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3133_one__reorient,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( one_on7727431528512463931l_num1 = X2 )
      = ( X2 = one_on7727431528512463931l_num1 ) ) ).

% one_reorient
thf(fact_3134_one__reorient,axiom,
    ! [X2: real] :
      ( ( one_one_real = X2 )
      = ( X2 = one_one_real ) ) ).

% one_reorient
thf(fact_3135_one__reorient,axiom,
    ! [X2: rat] :
      ( ( one_one_rat = X2 )
      = ( X2 = one_one_rat ) ) ).

% one_reorient
thf(fact_3136_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_3137_one__reorient,axiom,
    ! [X2: int] :
      ( ( one_one_int = X2 )
      = ( X2 = one_one_int ) ) ).

% one_reorient
thf(fact_3138_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_3139_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_3140_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_3141_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_3142_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3143_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3144_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3145_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3146_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3147_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3148_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3149_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3150_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3151_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3152_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3153_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3154_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_3155_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_3156_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_3157_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A @ B ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_3158_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_3159_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_3160_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_3161_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y2 ) @ one_one_nat ) ) ) ).

% insersimp'
thf(fact_3162_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L ) @ one_one_nat ) ) ) ).

% insertsimp'
thf(fact_3163_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_3164_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_3165_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_3166_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_3167_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_3168_add_Ogroup__left__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3169_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3170_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3171_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3172_add_Ocomm__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% add.comm_neutral
thf(fact_3173_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_3174_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_3175_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_3176_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_3177_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3178_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3179_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3180_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3181_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3182_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3183_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3184_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3185_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3186_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3187_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3188_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3189_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3190_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3191_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3192_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3193_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3194_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3195_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3196_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3197_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_3198_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3199_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3200_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3201_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_3202_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_3203_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3204_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3205_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3206_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_3207_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
        ? [C3: nat] :
          ( B4
          = ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_3208_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3209_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3210_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3211_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3212_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3213_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3214_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3215_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3216_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3217_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3218_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3219_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3220_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3221_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3222_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3223_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3224_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3225_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3226_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3227_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3228_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3229_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3230_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3231_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3232_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3233_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3234_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3235_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3236_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3237_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3238_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3239_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3240_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3241_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3242_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3243_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3244_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3245_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3246_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3247_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3248_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: uint32,Z4: uint32] : Y5 = Z4 )
    = ( ^ [A4: uint32,B4: uint32] :
          ( ( minus_minus_uint32 @ A4 @ B4 )
          = zero_zero_uint32 ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3249_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z4: real] : Y5 = Z4 )
    = ( ^ [A4: real,B4: real] :
          ( ( minus_minus_real @ A4 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3250_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z4: rat] : Y5 = Z4 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( minus_minus_rat @ A4 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3251_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
    = ( ^ [A4: int,B4: int] :
          ( ( minus_minus_int @ A4 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3252_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_3253_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_3254_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_3255_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_3256_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_3257_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_3258_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_3259_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_3260_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_3261_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3262_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3263_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3264_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_3265_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_3266_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_3267_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_3268_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_3269_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_3270_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_3271_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_3272_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_3273_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3274_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3275_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3276_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ one_on7727431528512463931l_num1 @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3277_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3278_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3279_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3280_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3281_mult_Ocomm__neutral,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A @ one_on7727431528512463931l_num1 )
      = A ) ).

% mult.comm_neutral
thf(fact_3282_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_3283_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_3284_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_3285_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_3286_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3287_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3288_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3289_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3290_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3291_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3292_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3293_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3294_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3295_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3296_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3297_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3298_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3299_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3300_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3301_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3302_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3303_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3304_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3305_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3306_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3307_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3308_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3309_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3310_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3311_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3312_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3313_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3314_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3315_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc2574133891255291104it_nat] :
      ( ! [Uu2: produc120671012495760973it_nat > produc120671012495760973it_nat > $o,Uv2: option2621746655072343315it_nat] :
          ( X2
         != ( produc5936680911947247184it_nat @ Uu2 @ ( 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 ) ) )
       => ~ ! [F2: produc120671012495760973it_nat > produc120671012495760973it_nat > $o,X3: produc120671012495760973it_nat,Y3: produc120671012495760973it_nat] :
              ( X2
             != ( produc5936680911947247184it_nat @ F2 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ X3 ) @ ( some_P2407035485129114418it_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3316_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc6000686143695694318it_nat] :
      ( ! [Uu2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > $o,Uv2: option7339022715339332451it_nat] :
          ( X2
         != ( produc3920266370798870110it_nat @ Uu2 @ ( 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 ) ) )
       => ~ ! [F2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > $o,X3: produc8047831477865546771it_nat,Y3: produc8047831477865546771it_nat] :
              ( X2
             != ( produc3920266370798870110it_nat @ F2 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ X3 ) @ ( some_P468703482102919278it_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3317_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X2
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X2
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3318_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X2
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X2
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X3: nat,Y3: nat] :
              ( X2
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3319_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X2
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X2
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X3: num,Y3: num] :
              ( X2
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3320_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc6872358179685758443it_nat] :
      ( ! [Uu2: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,Uv2: option2621746655072343315it_nat] :
          ( X2
         != ( produc8579712001971957723it_nat @ Uu2 @ ( 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 ) ) )
       => ~ ! [F2: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,A3: produc120671012495760973it_nat,B2: produc120671012495760973it_nat] :
              ( X2
             != ( produc8579712001971957723it_nat @ F2 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ A3 ) @ ( some_P2407035485129114418it_nat @ B2 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3321_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc5059602919146741221it_nat] :
      ( ! [Uu2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,Uv2: option7339022715339332451it_nat] :
          ( X2
         != ( produc2320005133921938071it_nat @ Uu2 @ ( 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 ) ) )
       => ~ ! [F2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,A3: produc8047831477865546771it_nat,B2: produc8047831477865546771it_nat] :
              ( X2
             != ( produc2320005133921938071it_nat @ F2 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ A3 ) @ ( some_P468703482102919278it_nat @ B2 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3322_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X2
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( X2
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3323_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X2
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X2
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A3: nat,B2: nat] :
              ( X2
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3324_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X2
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X2
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A3: num,B2: num] :
              ( X2
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B2 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3325_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% insert'_bound_height
thf(fact_3326_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_3327_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_3328_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_3329_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,Uv: option2621746655072343315it_nat] :
      ( ( vEBT_V819568868292977612it_nat @ Uu @ none_P1551326421579882414it_nat @ Uv )
      = none_P1551326421579882414it_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_3330_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,Uv: option7339022715339332451it_nat] :
      ( ( vEBT_V613753007643960916it_nat @ Uu @ none_P7668321371905463026it_nat @ Uv )
      = none_P7668321371905463026it_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_3331_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_3332_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: num > num > num,Uv: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_3333_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: nat > nat > nat,Uv: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_3334_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3335_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3336_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3337_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3338_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3339_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3340_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3341_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3342_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3343_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3344_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3345_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3346_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3347_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3348_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3349_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3350_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3351_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3352_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3353_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3354_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_3355_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3356_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3357_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_3358_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_3359_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_3360_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_3361_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_3362_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_3363_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_3364_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_3365_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_3366_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_3367_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_3368_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_3369_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_3370_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3371_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3372_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3373_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3374_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_3375_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3376_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3377_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3378_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3379_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3380_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3381_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3382_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3383_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3384_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3385_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3386_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3387_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3388_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3389_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3390_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3391_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3392_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3393_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3394_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3395_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_3396_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_3397_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3398_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_3399_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_3400_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3401_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3402_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3403_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3404_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3405_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3406_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3407_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_3408_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_3409_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3410_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3411_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3412_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3413_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3414_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3415_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3416_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3417_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3418_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3419_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3420_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3421_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3422_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3423_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,V: produc120671012495760973it_nat] :
      ( ( vEBT_V819568868292977612it_nat @ Uw @ ( some_P2407035485129114418it_nat @ V ) @ none_P1551326421579882414it_nat )
      = none_P1551326421579882414it_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_3424_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,V: produc8047831477865546771it_nat] :
      ( ( vEBT_V613753007643960916it_nat @ Uw @ ( some_P468703482102919278it_nat @ V ) @ none_P7668321371905463026it_nat )
      = none_P7668321371905463026it_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_3425_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_3426_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_3427_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_3428_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 ) ) )
         => ~ ! [A3: produc120671012495760973it_nat] :
                ( ( Xa
                  = ( some_P2407035485129114418it_nat @ A3 ) )
               => ! [B2: produc120671012495760973it_nat] :
                    ( ( Xb
                      = ( some_P2407035485129114418it_nat @ B2 ) )
                   => ( Y2
                     != ( some_P2407035485129114418it_nat @ ( X2 @ A3 @ B2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3429_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 ) ) )
         => ~ ! [A3: produc8047831477865546771it_nat] :
                ( ( Xa
                  = ( some_P468703482102919278it_nat @ A3 ) )
               => ! [B2: produc8047831477865546771it_nat] :
                    ( ( Xb
                      = ( some_P468703482102919278it_nat @ B2 ) )
                   => ( Y2
                     != ( some_P468703482102919278it_nat @ ( X2 @ A3 @ B2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3430_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 ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa
                  = ( some_P7363390416028606310at_nat @ A3 ) )
               => ! [B2: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B2 ) )
                   => ( Y2
                     != ( some_P7363390416028606310at_nat @ ( X2 @ A3 @ B2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3431_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 ) ) )
         => ~ ! [A3: num] :
                ( ( Xa
                  = ( some_num @ A3 ) )
               => ! [B2: num] :
                    ( ( Xb
                      = ( some_num @ B2 ) )
                   => ( Y2
                     != ( some_num @ ( X2 @ A3 @ B2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3432_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 ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa
                  = ( some_nat @ A3 ) )
               => ! [B2: nat] :
                    ( ( Xb
                      = ( some_nat @ B2 ) )
                   => ( Y2
                     != ( some_nat @ ( X2 @ A3 @ B2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3433_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_3434_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_3435_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3436_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3437_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3438_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3439_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3440_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3441_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_3442_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_3443_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3444_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3445_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3446_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3447_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3448_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3449_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3450_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3451_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3452_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3453_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3454_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3455_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3456_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3457_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A3: real,B2: real,C2: real] :
            ( ( P @ A3 @ B2 )
           => ( ( P @ B2 @ C2 )
             => ( ( ord_less_eq_real @ A3 @ B2 )
               => ( ( ord_less_eq_real @ B2 @ C2 )
                 => ( P @ A3 @ C2 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B )
               => ? [D4: real] :
                    ( ( ord_less_real @ zero_zero_real @ D4 )
                    & ! [A3: real,B2: real] :
                        ( ( ( ord_less_eq_real @ A3 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B2 )
                          & ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D4 ) )
                       => ( P @ A3 @ B2 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_3458_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_3459_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_3460_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_3461_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_3462_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_3463_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_3464_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_3465_vebt__predi_Osimps,axiom,
    ( vEBT_vebt_predi
    = ( ^ [T2: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
              @ ^ [Mima2: product_prod_nat_nat] :
                  ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                    @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                      @ ^ [L2: nat] :
                          ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                          @ ^ [H: nat] :
                              ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Minlow ) )
                                      @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_predi @ Aktnode @ L2 )
                                        @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                      @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_predi @ Summary3 @ H )
                                        @ ^ [Predsum: option_nat] :
                                            ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                            @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
          @ T2 ) ) ) ).

% vebt_predi.simps
thf(fact_3466_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ 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 @ T @ 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_3467_vebt__succi_Osimps,axiom,
    ( vEBT_vebt_succi
    = ( ^ [T2: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
              @ ^ [Mima2: product_prod_nat_nat] :
                  ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                    @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                        @ ^ [L2: nat] :
                            ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                            @ ^ [H: nat] :
                                ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Maxlow ) )
                                        @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_succi @ Aktnode @ L2 )
                                          @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                        @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_succi @ Summary3 @ H )
                                          @ ^ [Succsum: option_nat] :
                                              ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                              @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
          @ T2 ) ) ) ).

% vebt_succi.simps
thf(fact_3468_inrange,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_3469_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_3470_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_3471_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg ) )
     => ( ( vEBT_invar_vebt @ T @ Deg )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% height_double_log_univ_size
thf(fact_3472_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_3473_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V1232361888498592333_e_t_e @ T @ 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_3474_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] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_TBOUND_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
           => ( time_TBOUND_nat @ ( F3 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) )
       => ( time_TBOUND_nat @ ( vEBT_c1335663792808957512ap_nat @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_3475_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] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_T8353473612707095248on_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
           => ( time_T8353473612707095248on_nat @ ( F3 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) )
       => ( time_T8353473612707095248on_nat @ ( vEBT_c6250501799366334488on_nat @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_3476_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] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_3477_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] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_TBOUND_o @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
           => ( time_TBOUND_o @ ( F3 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) )
       => ( time_TBOUND_o @ ( vEBT_c6104975476656191286Heap_o @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_3478_subset__Collect__conv,axiom,
    ! [S2: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S2 @ ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ S2 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_3479_subset__Collect__conv,axiom,
    ! [S2: set_complex,P: complex > $o] :
      ( ( ord_le211207098394363844omplex @ S2 @ ( collect_complex @ P ) )
      = ( ! [X: complex] :
            ( ( member_complex @ X @ S2 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_3480_subset__Collect__conv,axiom,
    ! [S2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ S2 @ ( collec213857154873943460nt_int @ P ) )
      = ( ! [X: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X @ S2 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_3481_subset__Collect__conv,axiom,
    ! [S2: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ S2 @ ( collect_int @ P ) )
      = ( ! [X: int] :
            ( ( member_int @ X @ S2 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_3482_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F22: $o > $o > heap_T2636463487746394924on_nat,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6250501799366334488on_nat @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6250501799366334488on_nat
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3483_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T2636463487746394924on_nat > heap_T8145700208782473153_VEBTi,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F22: $o > $o > heap_T2636463487746394924on_nat,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6250501799366334488on_nat @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6028912655521741485_VEBTi
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3484_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T2636463487746394924on_nat > heap_Time_Heap_o,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F22: $o > $o > heap_T2636463487746394924on_nat,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6250501799366334488on_nat @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6104975476656191286Heap_o
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3485_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T8145700208782473153_VEBTi > heap_T2636463487746394924on_nat,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F22: $o > $o > heap_T8145700208782473153_VEBTi,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6028912655521741485_VEBTi @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6250501799366334488on_nat
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3486_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F22: $o > $o > heap_T8145700208782473153_VEBTi,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6028912655521741485_VEBTi @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6028912655521741485_VEBTi
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3487_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T8145700208782473153_VEBTi > heap_Time_Heap_o,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F22: $o > $o > heap_T8145700208782473153_VEBTi,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6028912655521741485_VEBTi @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6104975476656191286Heap_o
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3488_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,F22: $o > $o > heap_Time_Heap_o,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6104975476656191286Heap_o @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6250501799366334488on_nat
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3489_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,F22: $o > $o > heap_Time_Heap_o,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6104975476656191286Heap_o @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6028912655521741485_VEBTi
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3490_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,F22: $o > $o > heap_Time_Heap_o,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6104975476656191286Heap_o @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_c6104975476656191286Heap_o
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3491_VEBTi_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F22: $o > $o > heap_T2636463487746394924on_nat,X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_c6250501799366334488on_nat @ F1 @ F22 @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBTi.simps(5)
thf(fact_3492_VEBTi_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F22: $o > $o > heap_T8145700208782473153_VEBTi,X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_c6028912655521741485_VEBTi @ F1 @ F22 @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBTi.simps(5)
thf(fact_3493_VEBTi_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F22: $o > $o > heap_Time_Heap_o,X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_c6104975476656191286Heap_o @ F1 @ F22 @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBTi.simps(5)
thf(fact_3494_VEBTi_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F22: $o > $o > heap_T2636463487746394924on_nat,X21: $o,X22: $o] :
      ( ( vEBT_c6250501799366334488on_nat @ F1 @ F22 @ ( vEBT_Leafi @ X21 @ X22 ) )
      = ( F22 @ X21 @ X22 ) ) ).

% VEBTi.simps(6)
thf(fact_3495_VEBTi_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F22: $o > $o > heap_T8145700208782473153_VEBTi,X21: $o,X22: $o] :
      ( ( vEBT_c6028912655521741485_VEBTi @ F1 @ F22 @ ( vEBT_Leafi @ X21 @ X22 ) )
      = ( F22 @ X21 @ X22 ) ) ).

% VEBTi.simps(6)
thf(fact_3496_VEBTi_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F22: $o > $o > heap_Time_Heap_o,X21: $o,X22: $o] :
      ( ( vEBT_c6104975476656191286Heap_o @ F1 @ F22 @ ( vEBT_Leafi @ X21 @ X22 ) )
      = ( F22 @ X21 @ X22 ) ) ).

% VEBTi.simps(6)
thf(fact_3497_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3498_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3499_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3500_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3501_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3502_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3503_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3504_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3505_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3506_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3507_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3508_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3509_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3510_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3511_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3512_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3513_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3514_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3515_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3516_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3517_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ~ ( ord_less_real @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3518_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ~ ( ord_less_rat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3519_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ~ ( ord_less_num @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3520_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_nat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3521_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ~ ( ord_less_int @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3522_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ord_less_real @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3523_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ord_less_rat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3524_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( ord_less_num @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3525_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_nat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3526_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ord_less_int @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3527_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3528_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3529_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3530_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3531_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3532_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3533_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3534_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3535_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3536_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q7: 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 )
              = ( Q7 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q7 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3537_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3538_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3539_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3540_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3541_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3542_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3543_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3544_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3545_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3546_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3547_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ord_less_real @ X6 @ T ) ) ).

% minf(5)
thf(fact_3548_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ord_less_rat @ X6 @ T ) ) ).

% minf(5)
thf(fact_3549_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( ord_less_num @ X6 @ T ) ) ).

% minf(5)
thf(fact_3550_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_nat @ X6 @ T ) ) ).

% minf(5)
thf(fact_3551_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ord_less_int @ X6 @ T ) ) ).

% minf(5)
thf(fact_3552_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ~ ( ord_less_real @ T @ X6 ) ) ).

% minf(7)
thf(fact_3553_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ~ ( ord_less_rat @ T @ X6 ) ) ).

% minf(7)
thf(fact_3554_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ~ ( ord_less_num @ T @ X6 ) ) ).

% minf(7)
thf(fact_3555_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_nat @ T @ X6 ) ) ).

% minf(7)
thf(fact_3556_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ~ ( ord_less_int @ T @ X6 ) ) ).

% minf(7)
thf(fact_3557_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ~ ( ord_less_eq_real @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3558_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ~ ( ord_less_eq_rat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3559_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ~ ( ord_less_eq_num @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3560_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3561_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ~ ( ord_less_eq_int @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3562_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ord_less_eq_real @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3563_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ord_less_eq_rat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3564_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( ord_less_eq_num @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3565_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3566_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ord_less_eq_int @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3567_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ord_less_eq_real @ X6 @ T ) ) ).

% minf(6)
thf(fact_3568_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ord_less_eq_rat @ X6 @ T ) ) ).

% minf(6)
thf(fact_3569_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( ord_less_eq_num @ X6 @ T ) ) ).

% minf(6)
thf(fact_3570_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_3571_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ord_less_eq_int @ X6 @ T ) ) ).

% minf(6)
thf(fact_3572_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X6 ) ) ).

% minf(8)
thf(fact_3573_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X6 ) ) ).

% minf(8)
thf(fact_3574_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X6 ) ) ).

% minf(8)
thf(fact_3575_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_3576_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X6 ) ) ).

% minf(8)
thf(fact_3577_inf__period_I1_J,axiom,
    ! [P: real > $o,D5: real,Q: real > $o] :
      ( ! [X3: real,K3: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D5 ) ) ) )
     => ( ! [X3: real,K3: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D5 ) ) ) )
       => ! [X6: real,K5: real] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K5 @ D5 ) ) )
              & ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K5 @ D5 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3578_inf__period_I1_J,axiom,
    ! [P: rat > $o,D5: rat,Q: rat > $o] :
      ( ! [X3: rat,K3: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D5 ) ) ) )
     => ( ! [X3: rat,K3: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D5 ) ) ) )
       => ! [X6: rat,K5: rat] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K5 @ D5 ) ) )
              & ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K5 @ D5 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3579_inf__period_I1_J,axiom,
    ! [P: int > $o,D5: int,Q: int > $o] :
      ( ! [X3: int,K3: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D5 ) ) ) )
     => ( ! [X3: int,K3: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D5 ) ) ) )
       => ! [X6: int,K5: int] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K5 @ D5 ) ) )
              & ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K5 @ D5 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3580_inf__period_I2_J,axiom,
    ! [P: real > $o,D5: real,Q: real > $o] :
      ( ! [X3: real,K3: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D5 ) ) ) )
     => ( ! [X3: real,K3: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D5 ) ) ) )
       => ! [X6: real,K5: real] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K5 @ D5 ) ) )
              | ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K5 @ D5 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3581_inf__period_I2_J,axiom,
    ! [P: rat > $o,D5: rat,Q: rat > $o] :
      ( ! [X3: rat,K3: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D5 ) ) ) )
     => ( ! [X3: rat,K3: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D5 ) ) ) )
       => ! [X6: rat,K5: rat] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K5 @ D5 ) ) )
              | ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K5 @ D5 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3582_inf__period_I2_J,axiom,
    ! [P: int > $o,D5: int,Q: int > $o] :
      ( ! [X3: int,K3: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D5 ) ) ) )
     => ( ! [X3: int,K3: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D5 ) ) ) )
       => ! [X6: int,K5: int] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K5 @ D5 ) ) )
              | ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K5 @ D5 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3583_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_3584_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_3585_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_3586_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d2 @ T @ 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_3587_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c2 @ T @ 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_3588_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ 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_3589_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A @ B ) )
      = one_one_real ) ).

% VEBT_internal.cnt.simps(1)
thf(fact_3590_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Leaf @ A @ B ) )
      = one_one_nat ) ).

% VEBT_internal.cnt'.simps(1)
thf(fact_3591_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_1: int] : ( P6 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_3592_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_3593_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 )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3594_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 )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3595_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_3596_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_3597_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A @ B ) )
      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.space'.simps(1)
thf(fact_3598_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A @ B ) )
      = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.space.simps(1)
thf(fact_3599_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B ) )
          = ( semiri5074537144036343181t_real @ B ) ) ) ) ).

% log_pow_cancel
thf(fact_3600_log__le__cancel__iff,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) )
            = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_3601_log__le__one__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ one_one_real )
          = ( ord_less_eq_real @ X2 @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_3602_one__le__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X2 ) )
          = ( ord_less_eq_real @ A @ X2 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_3603_log__le__zero__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_3604_zero__le__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X2 ) )
          = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_3605_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_3606_log__eq__one,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ A )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_3607_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_3608_zero__less__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X2 ) )
          = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_3609_log__less__zero__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A @ X2 ) @ zero_zero_real )
          = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_3610_one__less__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X2 ) )
          = ( ord_less_real @ A @ X2 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_3611_log__less__one__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A @ X2 ) @ one_one_real )
          = ( ord_less_real @ X2 @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_3612_log__less__cancel__iff,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) )
            = ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_3613_log__base__change,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X2 )
          = ( divide_divide_real @ ( log @ A @ X2 ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_3614_less__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_3615_log__of__power__eq,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N2 )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_3616_log__mult,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y2 )
           => ( ( log @ A @ ( times_times_real @ X2 @ Y2 ) )
              = ( plus_plus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_3617_le__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_3618_log__divide,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y2 )
           => ( ( log @ A @ ( divide_divide_real @ X2 @ Y2 ) )
              = ( minus_minus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_3619_log__base__pow,axiom,
    ! [A: real,N2: nat,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N2 ) @ X2 )
        = ( divide_divide_real @ ( log @ A @ X2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_3620_log__nat__power,axiom,
    ! [X2: real,B: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ B @ ( power_power_real @ X2 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X2 ) ) ) ) ).

% log_nat_power
thf(fact_3621_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_3622_log__of__power__less,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_3623_log__of__power__le,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_3624_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_3625_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_3626_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_3627_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3628_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3629_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3630_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3631_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3632_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3633_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3634_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3635_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X2 ) )
       => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_3636_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_3637_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) )
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_3638_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y2 )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y2 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( Y2
                  = ( ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) )
                 => ( 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 @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( Y2
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_3639_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N4: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_3640_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N4: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_3641_del__single__cont,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( X2 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_3642_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X2 ) @ N2 ) ) ).

% delete_pres_valid
thf(fact_3643_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_3644_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X2 ) @ Y2 )
        = ( ( X2 != Y2 )
          & ( vEBT_V8194947554948674370ptions @ T @ Y2 ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_3645_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X2 ) @ Y2 )
        = ( ( X2 != Y2 )
          & ( vEBT_vebt_member @ T @ Y2 ) ) ) ) ).

% dele_member_cont_corr
thf(fact_3646_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X2 ) )
       => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X2 ) @ one_one_nat ) ) ) ).

% minNull_delete_time_bound'
thf(fact_3647_Icc__eq__Icc,axiom,
    ! [L: set_int,H2: set_int,L4: set_int,H4: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H2 )
        = ( set_or370866239135849197et_int @ L4 @ H4 ) )
      = ( ( ( L = L4 )
          & ( H2 = H4 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H2 )
          & ~ ( ord_less_eq_set_int @ L4 @ H4 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3648_Icc__eq__Icc,axiom,
    ! [L: rat,H2: rat,L4: rat,H4: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H2 )
        = ( set_or633870826150836451st_rat @ L4 @ H4 ) )
      = ( ( ( L = L4 )
          & ( H2 = H4 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H2 )
          & ~ ( ord_less_eq_rat @ L4 @ H4 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3649_Icc__eq__Icc,axiom,
    ! [L: num,H2: num,L4: num,H4: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H2 )
        = ( set_or7049704709247886629st_num @ L4 @ H4 ) )
      = ( ( ( L = L4 )
          & ( H2 = H4 ) )
        | ( ~ ( ord_less_eq_num @ L @ H2 )
          & ~ ( ord_less_eq_num @ L4 @ H4 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3650_Icc__eq__Icc,axiom,
    ! [L: nat,H2: nat,L4: nat,H4: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
        = ( set_or1269000886237332187st_nat @ L4 @ H4 ) )
      = ( ( ( L = L4 )
          & ( H2 = H4 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H2 )
          & ~ ( ord_less_eq_nat @ L4 @ H4 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3651_Icc__eq__Icc,axiom,
    ! [L: int,H2: int,L4: int,H4: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H2 )
        = ( set_or1266510415728281911st_int @ L4 @ H4 ) )
      = ( ( ( L = L4 )
          & ( H2 = H4 ) )
        | ( ~ ( ord_less_eq_int @ L @ H2 )
          & ~ ( ord_less_eq_int @ L4 @ H4 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3652_Icc__eq__Icc,axiom,
    ! [L: code_integer,H2: code_integer,L4: code_integer,H4: code_integer] :
      ( ( ( set_or189985376899183464nteger @ L @ H2 )
        = ( set_or189985376899183464nteger @ L4 @ H4 ) )
      = ( ( ( L = L4 )
          & ( H2 = H4 ) )
        | ( ~ ( ord_le3102999989581377725nteger @ L @ H2 )
          & ~ ( ord_le3102999989581377725nteger @ L4 @ H4 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3653_Icc__eq__Icc,axiom,
    ! [L: real,H2: real,L4: real,H4: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H2 )
        = ( set_or1222579329274155063t_real @ L4 @ H4 ) )
      = ( ( ( L = L4 )
          & ( H2 = H4 ) )
        | ( ~ ( ord_less_eq_real @ L @ H2 )
          & ~ ( ord_less_eq_real @ L4 @ H4 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3654_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I )
        & ( ord_less_eq_set_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3655_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3656_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3657_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3658_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3659_atLeastAtMost__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or189985376899183464nteger @ L @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L @ I )
        & ( ord_le3102999989581377725nteger @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3660_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3661_delt__out__of__range,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
        | ( ord_less_nat @ Ma @ X2 ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_3662_aset_I2_J,axiom,
    ! [D5: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D5 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D5 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
             => ( ( P @ ( plus_plus_int @ X6 @ D5 ) )
                | ( Q @ ( plus_plus_int @ X6 @ D5 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_3663_aset_I1_J,axiom,
    ! [D5: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D5 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D5 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
             => ( ( P @ ( plus_plus_int @ X6 @ D5 ) )
                & ( Q @ ( plus_plus_int @ X6 @ D5 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_3664_bset_I2_J,axiom,
    ! [D5: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D5 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D5 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
             => ( ( P @ ( minus_minus_int @ X6 @ D5 ) )
                | ( Q @ ( minus_minus_int @ X6 @ D5 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_3665_bset_I1_J,axiom,
    ! [D5: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D5 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D5 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
             => ( ( P @ ( minus_minus_int @ X6 @ D5 ) )
                & ( Q @ ( minus_minus_int @ X6 @ D5 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_3666_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M8: nat] :
      ( ( P @ X2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M8 ) )
       => ~ ! [M2: nat] :
              ( ( P @ M2 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_3667_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_3668_aset_I7_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X6 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X6 @ D5 ) ) ) ) ) ).

% aset(7)
thf(fact_3669_aset_I5_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ T @ A2 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X6 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X6 @ D5 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_3670_aset_I4_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ T @ A2 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( plus_plus_int @ X6 @ D5 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_3671_aset_I3_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( plus_plus_int @ X6 @ D5 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_3672_bset_I7_J,axiom,
    ! [D5: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ T @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X6 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X6 @ D5 ) ) ) ) ) ) ).

% bset(7)
thf(fact_3673_bset_I5_J,axiom,
    ! [D5: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X6 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X6 @ D5 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_3674_bset_I4_J,axiom,
    ! [D5: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ T @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( minus_minus_int @ X6 @ D5 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_3675_bset_I3_J,axiom,
    ! [D5: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( minus_minus_int @ X6 @ D5 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_3676_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ( ? [X5: int] : ( P @ X5 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_3677_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_3678_aset_I8_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X6 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X6 @ D5 ) ) ) ) ) ).

% aset(8)
thf(fact_3679_aset_I6_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X6 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X6 @ D5 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_3680_bset_I8_J,axiom,
    ! [D5: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X6 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X6 @ D5 ) ) ) ) ) ) ).

% bset(8)
thf(fact_3681_bset_I6_J,axiom,
    ! [D5: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X6 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X6 @ D5 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_3682_cpmi,axiom,
    ! [D5: int,P: int > $o,P6: int > $o,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B5 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D5 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D5 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ B5 )
                        & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_3683_cppi,axiom,
    ! [D5: int,P: int > $o,P6: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D5 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A2 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D5 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D5 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ A2 )
                        & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3684_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N4: nat] :
            ( ~ ( P @ N4 )
            & ( P @ ( suc @ N4 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_3685_real__arch__simple,axiom,
    ! [X2: real] :
    ? [N4: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% real_arch_simple
thf(fact_3686_real__arch__simple,axiom,
    ! [X2: rat] :
    ? [N4: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% real_arch_simple
thf(fact_3687_reals__Archimedean2,axiom,
    ! [X2: rat] :
    ? [N4: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% reals_Archimedean2
thf(fact_3688_reals__Archimedean2,axiom,
    ! [X2: real] :
    ? [N4: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% reals_Archimedean2
thf(fact_3689_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B @ D )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3690_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3691_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3692_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3693_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3694_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A @ B )
          | ( ( ord_le3102999989581377725nteger @ C @ A )
            & ( ord_le3102999989581377725nteger @ B @ D )
            & ( ( ord_le6747313008572928689nteger @ C @ A )
              | ( ord_le6747313008572928689nteger @ B @ D ) ) ) )
        & ( ord_le3102999989581377725nteger @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3695_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3696_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_3697_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_3698_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_3699_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ 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 @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_3700_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_3701_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_3702_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) )
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_3703_del__x__mia,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_3704_del__x__mi__lets__in__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_3705_del__x__mi__lets__in,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_3706_del__x__mi,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_3707_del__in__range,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_3708_del__x__not__mia,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X2 = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_3709_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_3710_del__x__not__mi,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_3711_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( ~ ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ J @ X2 ) @ I )
        = ( nth_VEBT_VEBT @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_3712_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,X2: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ J @ X2 ) @ I )
        = ( nth_VEBT_VEBTi @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_3713_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,X2: real] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( nth_real @ ( list_update_real @ L @ J @ X2 ) @ I )
        = ( nth_real @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_3714_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J: nat,X2: $o] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( nth_o @ ( list_update_o @ L @ J @ X2 ) @ I )
        = ( nth_o @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_3715_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,X2: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( list_update_nat @ L @ J @ X2 ) @ I )
        = ( nth_nat @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_3716_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,X2: int] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( nth_int @ ( list_update_int @ L @ J @ X2 ) @ I )
        = ( nth_int @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_3717_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_3718_del__x__mi__lets__in__not__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_3719_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
     => ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ L ) )
            & ! [Y: produc6575502325842934193n_assn] : ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3720_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L ) )
            & ! [Y: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3721_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi,Y2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L ) )
            & ! [Y: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3722_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,X2: real,Y2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_real @ X2 @ ( set_real2 @ L ) )
            & ! [Y: real] : ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3723_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,X2: $o,Y2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_o @ X2 @ ( set_o2 @ L ) )
            & ! [Y: $o] : ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3724_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_nat @ X2 @ ( set_nat2 @ L ) )
            & ! [Y: nat] : ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3725_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X2: int,Y2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_int @ X2 @ ( set_int2 @ L ) )
            & ! [Y: int] : ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3726_in__set__upd__cases,axiom,
    ! [X2: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn,I: nat,Y2: produc6575502325842934193n_assn] :
      ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
         => ( X2 != Y2 ) )
       => ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_3727_in__set__upd__cases,axiom,
    ! [X2: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
         => ( X2 != Y2 ) )
       => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_3728_in__set__upd__cases,axiom,
    ! [X2: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y2: vEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
         => ( X2 != Y2 ) )
       => ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_3729_in__set__upd__cases,axiom,
    ! [X2: real,L: list_real,I: nat,Y2: real] :
      ( ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
         => ( X2 != Y2 ) )
       => ( member_real @ X2 @ ( set_real2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_3730_in__set__upd__cases,axiom,
    ! [X2: $o,L: list_o,I: nat,Y2: $o] :
      ( ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
         => ( X2 = ~ Y2 ) )
       => ( member_o @ X2 @ ( set_o2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_3731_in__set__upd__cases,axiom,
    ! [X2: nat,L: list_nat,I: nat,Y2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
         => ( X2 != Y2 ) )
       => ( member_nat @ X2 @ ( set_nat2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_3732_in__set__upd__cases,axiom,
    ! [X2: int,L: list_int,I: nat,Y2: int] :
      ( ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
         => ( X2 != Y2 ) )
       => ( member_int @ X2 @ ( set_int2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_3733_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
     => ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: produc6575502325842934193n_assn] : ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3734_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3735_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi,Y2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3736_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,X2: real,Y2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: real] : ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3737_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,X2: $o,Y2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: $o] : ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3738_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: nat] : ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3739_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X2: int,Y2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: int] : ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3740_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X2 ) @ J )
          = ( nth_VEBT_VEBT @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3741_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X2 ) @ J )
          = ( nth_VEBT_VEBTi @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3742_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_real,X2: real] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X2 ) @ J )
          = ( nth_real @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3743_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,X2: $o,J: nat] :
      ( ( nth_o @ ( list_update_o @ L @ I @ X2 ) @ J )
      = ( ( ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => X2 )
        & ( ~ ( ( I = J )
              & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => ( nth_o @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3744_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_nat,X2: nat] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X2 ) @ J )
          = ( nth_nat @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3745_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_int,X2: int] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X2 ) @ J )
          = ( nth_int @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3746_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,Va: nat,TreeList: list_VEBT_VEBT,Summary: 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 @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ TreeList @ Summary ) @ X2 )
              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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_3747_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 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A3: $o,B2: $o] :
                ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ? [N4: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N4 ) ) )
               => ( Y2 != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y2 != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y2 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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_3748_vebt__delete_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( Xa = zero_zero_nat )
             => ( Y2
               != ( vEBT_Leaf @ $false @ B2 ) ) ) )
       => ( ! [A3: $o] :
              ( ? [B2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ( Y2
                 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ? [N4: nat] :
                      ( Xa
                      = ( suc @ ( suc @ N4 ) ) )
                 => ( Y2
                   != ( vEBT_Leaf @ A3 @ B2 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( Y2
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                   => ( Y2
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                     => ( Y2
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( 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 ) ) @ TreeList2 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                                & ( ~ ( ( 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_3749_vebt__delete_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X2 @ Mi )
          | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X2 @ Mi )
            | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_3750_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_3751_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_3752_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_3753_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_3754_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_3755_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_3756_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_3757_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ X2 @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_3758_insert__simp__norm,axiom,
    ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ Mi @ X2 )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_3759_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_3760_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_3761_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_3762_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_3763_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_3764_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_3765_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_3766_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X2 ) ).

% buildup_nothing_in_leaf
thf(fact_3767_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X )
          | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_3768_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_3769_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_3770_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_3771_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_3772_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_3773_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_3774_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3775_list__update__id,axiom,
    ! [Xs2: list_int,I: nat] :
      ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3776_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_3777_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_3778_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_3779_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_3780_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_3781_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_3782_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_3783_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_3784_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_3785_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_3786_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_3787_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_3788_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_3789_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_3790_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_3791_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_3792_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_3793_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_3794_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_3795_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_3796_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_3797_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_3798_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_3799_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_3800_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_3801_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_3802_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_3803_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_3804_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_3805_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_3806_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_3807_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_3808_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_3809_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_3810_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_3811_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_3812_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_3813_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_3814_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_3815_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_3816_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_3817_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_3818_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_3819_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_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_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_3827_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_3828_max__add__distrib__right,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( plus_plus_real @ X2 @ ( ord_max_real @ Y2 @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( plus_plus_real @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3829_max__add__distrib__right,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( plus_plus_rat @ X2 @ ( ord_max_rat @ Y2 @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( plus_plus_rat @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3830_max__add__distrib__right,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y2 @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3831_max__add__distrib__right,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( plus_plus_int @ X2 @ ( ord_max_int @ Y2 @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( plus_plus_int @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3832_max__add__distrib__left,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X2 @ Y2 ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X2 @ Z ) @ ( plus_plus_real @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3833_max__add__distrib__left,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X2 @ Y2 ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Z ) @ ( plus_plus_rat @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3834_max__add__distrib__left,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y2 ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3835_max__add__distrib__left,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X2 @ Y2 ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X2 @ Z ) @ ( plus_plus_int @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3836_max__diff__distrib__left,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X2 @ Y2 ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X2 @ Z ) @ ( minus_minus_real @ Y2 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3837_max__diff__distrib__left,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X2 @ Y2 ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X2 @ Z ) @ ( minus_minus_rat @ Y2 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3838_max__diff__distrib__left,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X2 @ Y2 ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X2 @ Z ) @ ( minus_minus_int @ Y2 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3839_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_3840_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_3841_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_3842_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_3843_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_3844_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_3845_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_3846_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_3847_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_3848_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_3849_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_real] :
      ( ( size_size_list_real @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3850_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3851_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3852_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3853_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_3854_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_3855_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_3856_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_3857_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_3858_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_3859_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_3860_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_3861_subset__code_I1_J,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,B5: set_Pr5949110396991348497n_assn] :
      ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs2 ) @ B5 )
      = ( ! [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
           => ( member7957490590177025114n_assn @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_3862_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_3863_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : Y5 = Z4 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
                = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3864_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBTi,Z4: list_VEBT_VEBTi] : Y5 = Z4 )
    = ( ^ [Xs: list_VEBT_VEBTi,Ys3: list_VEBT_VEBTi] :
          ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
             => ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                = ( nth_VEBT_VEBTi @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3865_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_real,Z4: list_real] : Y5 = Z4 )
    = ( ^ [Xs: list_real,Ys3: list_real] :
          ( ( ( size_size_list_real @ Xs )
            = ( size_size_list_real @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
             => ( ( nth_real @ Xs @ I2 )
                = ( nth_real @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3866_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z4: list_o] : Y5 = Z4 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I2 )
                = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3867_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z4: list_nat] : Y5 = Z4 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I2 )
                = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3868_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z4: list_int] : Y5 = Z4 )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I2 )
                = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3869_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: vEBT_VEBT] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3870_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBTi > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: vEBT_VEBTi] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3871_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > real > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: real] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list_real] :
            ( ( ( size_size_list_real @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_real @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3872_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: $o] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_o @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3873_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: nat] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3874_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: int] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_int @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3875_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3876_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
        = ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( ( nth_VEBT_VEBTi @ Xs2 @ I3 )
              = ( nth_VEBT_VEBTi @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3877_nth__equalityI,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
           => ( ( nth_real @ Xs2 @ I3 )
              = ( nth_real @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3878_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I3 )
              = ( nth_o @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3879_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I3 )
              = ( nth_nat @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3880_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I3 )
              = ( nth_int @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3881_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M7: nat] :
            ( ( ord_less_eq_nat @ M7 @ N2 )
           => ( P @ M7 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_3882_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M7: nat] :
            ( ( ord_less_eq_nat @ M7 @ N2 )
            & ( P @ M7 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_3883_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A2: set_nat,X2: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
     => ( ( member_nat @ X2 @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3884_set__update__subsetI,axiom,
    ! [Xs2: list_real,A2: set_real,X2: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
     => ( ( member_real @ X2 @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3885_set__update__subsetI,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,A2: set_Pr5949110396991348497n_assn,X2: produc6575502325842934193n_assn,I: nat] :
      ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs2 ) @ A2 )
     => ( ( member7957490590177025114n_assn @ X2 @ A2 )
       => ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3886_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X2: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
     => ( ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3887_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBTi,A2: set_VEBT_VEBTi,X2: vEBT_VEBTi,I: nat] :
      ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs2 ) @ A2 )
     => ( ( member_VEBT_VEBTi @ X2 @ A2 )
       => ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3888_set__update__subsetI,axiom,
    ! [Xs2: list_int,A2: set_int,X2: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
     => ( ( member_int @ X2 @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3889_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N2 ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_3890_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_3891_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_3892_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_3893_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_3894_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_3895_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_3896_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_3897_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 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3898_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 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3899_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 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
           => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3900_all__set__conv__all__nth,axiom,
    ! [Xs2: list_real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
           => ( P @ ( nth_real @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3901_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3902_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3903_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3904_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,X2: vEBT_VEBTi] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I3 ) ) )
     => ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3905_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3906_all__nth__imp__all__set,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o,X2: produc6575502325842934193n_assn] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
         => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ I3 ) ) )
     => ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3907_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X2: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
     => ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3908_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X2: $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
     => ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3909_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X2: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3910_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X2: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
     => ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3911_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
            & ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3912_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3913_in__set__conv__nth,axiom,
    ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
            & ( ( nth_Pr1769885009046257848n_assn @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3914_in__set__conv__nth,axiom,
    ! [X2: real,Xs2: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3915_in__set__conv__nth,axiom,
    ! [X2: $o,Xs2: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3916_in__set__conv__nth,axiom,
    ! [X2: nat,Xs2: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3917_in__set__conv__nth,axiom,
    ! [X2: int,Xs2: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3918_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_3919_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_3920_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_3921_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_3922_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_3923_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_3924_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_3925_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_3926_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_3927_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_3928_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_3929_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_3930_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_3931_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_3932_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_3933_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_3934_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_3935_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_3936_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_3937_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_3938_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_3939_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_3940_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_3941_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_3942_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_3943_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_3944_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_3945_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_3946_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_3947_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_3948_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_3949_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_3950_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_3951_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_3952_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_3953_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S3 ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_3954_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) ).

% vebt_delete.simps(5)
thf(fact_3955_vebt__insert_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y2
                    = ( vEBT_Leaf @ $true @ B2 ) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y2
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y2
                        = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
             => ( Y2
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
               => ( Y2
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                 => ( Y2
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( 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 @ TreeList2 ) )
                          & ~ ( ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_3956_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y2
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y2 )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( Y2
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_3957_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A3 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B2 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_3958_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A3 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B2 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_3959_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) ).

% vebt_delete.simps(6)
thf(fact_3960_httI,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_option_nat @ C @ H3 ) @ T ) )
       => ( time_htt_option_nat @ P @ C @ Q @ T ) ) ) ).

% httI
thf(fact_3961_httI,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_nat @ C @ H3 ) @ T ) )
       => ( time_htt_nat @ P @ C @ Q @ T ) ) ) ).

% httI
thf(fact_3962_httI,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_o @ C @ H3 ) @ T ) )
       => ( time_htt_o @ P @ C @ Q @ T ) ) ) ).

% httI
thf(fact_3963_httI,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ C @ H3 ) @ T ) )
       => ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T ) ) ) ).

% httI
thf(fact_3964_httI,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn,T: nat] :
      ( ( hoare_8945653483474564448t_unit @ P @ C @ Q )
     => ( ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ C @ H3 ) @ T ) )
       => ( time_h7375078312994015913t_unit @ P @ C @ Q @ T ) ) ) ).

% httI
thf(fact_3965_htt__def,axiom,
    ( time_htt_option_nat
    = ( ^ [P3: assn,C3: heap_T2636463487746394924on_nat,Q4: option_nat > assn,T2: nat] :
          ( ( hoare_7629718768684598413on_nat @ P3 @ C3 @ Q4 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_option_nat @ C3 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_3966_htt__def,axiom,
    ( time_htt_nat
    = ( ^ [P3: assn,C3: heap_Time_Heap_nat,Q4: nat > assn,T2: nat] :
          ( ( hoare_3067605981109127869le_nat @ P3 @ C3 @ Q4 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_nat @ C3 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_3967_htt__def,axiom,
    ( time_htt_o
    = ( ^ [P3: assn,C3: heap_Time_Heap_o,Q4: $o > assn,T2: nat] :
          ( ( hoare_hoare_triple_o @ P3 @ C3 @ Q4 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_o @ C3 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_3968_htt__def,axiom,
    ( time_htt_VEBT_VEBTi
    = ( ^ [P3: assn,C3: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,T2: nat] :
          ( ( hoare_1429296392585015714_VEBTi @ P3 @ C3 @ Q4 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ C3 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_3969_htt__def,axiom,
    ( time_h7375078312994015913t_unit
    = ( ^ [P3: assn,C3: heap_T5738788834812785303t_unit,Q4: product_unit > assn,T2: nat] :
          ( ( hoare_8945653483474564448t_unit @ P3 @ C3 @ Q4 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ C3 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_3970_succ__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T @ Y )
                & ( ord_less_nat @ X2 @ Y ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_3971_pred__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T @ Y )
                & ( ord_less_nat @ Y @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_3972_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 ) )
       => ( ! [Uu2: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( B2
                     => ( Y2
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B2
                     => ( Y2 = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B2 ) @ 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 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2 = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd2 ) )
                   => ( ( Y2 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( 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 @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_3973_psubsetI,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( A2 != B5 )
       => ( ord_less_set_int @ A2 @ B5 ) ) ) ).

% psubsetI
thf(fact_3974_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_3975_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_3976_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y23: num] :
      ( ( ( bit0 @ X23 )
        = ( bit0 @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% verit_eq_simplify(8)
thf(fact_3977_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3978_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3979_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3980_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3981_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3982_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3983_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_3984_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_3985_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_3986_subset__antisym,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( A2 = B5 ) ) ) ).

% subset_antisym
thf(fact_3987_subsetI,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( member_nat @ X3 @ B5 ) )
     => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3988_subsetI,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( member_VEBT_VEBT @ X3 @ B5 ) )
     => ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3989_subsetI,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( member_real @ X3 @ B5 ) )
     => ( ord_less_eq_set_real @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3990_subsetI,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( member_int @ X3 @ B5 ) )
     => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3991_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_3992_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_3993_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B5 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_3994_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B5 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_3995_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B5 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_3996_subset__empty,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_3997_subset__empty,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_3998_subset__empty,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_3999_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_4000_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_4001_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_4002_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4003_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4004_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4005_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4006_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4007_atLeastatMost__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or189985376899183464nteger @ A @ B ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4008_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4009_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4010_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4011_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4012_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4013_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4014_atLeastatMost__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4015_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4016_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_4017_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_4018_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_4019_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_4020_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_4021_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_4022_mod__pure,axiom,
    ! [B: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( pure_assn @ B ) @ H2 )
      = ( ( ( produc8586169260539613262et_nat @ H2 )
          = bot_bot_set_nat )
        & B ) ) ).

% mod_pure
thf(fact_4023_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_4024_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = B ) ).

% mod_h_bot_iff(1)
thf(fact_4025_mod__h__bot__iff_I4_J,axiom,
    ! [Q2: array_VEBT_VEBTi,Y2: list_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ~ ( rep_assn @ ( snga_assn_VEBT_VEBTi @ Q2 @ Y2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(4)
thf(fact_4026_mod__h__bot__iff_I8_J,axiom,
    ! [R2: list_VEBT_VEBTi > assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ R2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ? [X: list_VEBT_VEBTi] : ( rep_assn @ ( R2 @ X ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(8)
thf(fact_4027_Set_Oempty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X: complex] : $false ) ) ).

% Set.empty_def
thf(fact_4028_Set_Oempty__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X: product_prod_int_int] : $false ) ) ).

% Set.empty_def
thf(fact_4029_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_4030_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X: int] : $false ) ) ).

% Set.empty_def
thf(fact_4031_Set_Oempty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X: real] : $false ) ) ).

% Set.empty_def
thf(fact_4032_set__notEmptyE,axiom,
    ! [S2: set_VEBT_VEBT] :
      ( ( S2 != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X3: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X3 @ S2 ) ) ).

% set_notEmptyE
thf(fact_4033_set__notEmptyE,axiom,
    ! [S2: set_nat] :
      ( ( S2 != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S2 ) ) ).

% set_notEmptyE
thf(fact_4034_set__notEmptyE,axiom,
    ! [S2: set_int] :
      ( ( S2 != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S2 ) ) ).

% set_notEmptyE
thf(fact_4035_set__notEmptyE,axiom,
    ! [S2: set_real] :
      ( ( S2 != bot_bot_set_real )
     => ~ ! [X3: real] :
            ~ ( member_real @ X3 @ S2 ) ) ).

% set_notEmptyE
thf(fact_4036_memb__imp__not__empty,axiom,
    ! [X2: vEBT_VEBT,S2: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ S2 )
     => ( S2 != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
thf(fact_4037_memb__imp__not__empty,axiom,
    ! [X2: nat,S2: set_nat] :
      ( ( member_nat @ X2 @ S2 )
     => ( S2 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_4038_memb__imp__not__empty,axiom,
    ! [X2: int,S2: set_int] :
      ( ( member_int @ X2 @ S2 )
     => ( S2 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
thf(fact_4039_memb__imp__not__empty,axiom,
    ! [X2: real,S2: set_real] :
      ( ( member_real @ X2 @ S2 )
     => ( S2 != bot_bot_set_real ) ) ).

% memb_imp_not_empty
thf(fact_4040_times__assn__raw_Ocases,axiom,
    ! [X2: produc2732055786443039994et_nat] :
      ~ ! [P7: produc3658429121746597890et_nat > $o,Q8: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X2
         != ( produc2245416461498447860et_nat @ P7 @ ( produc5001842942810119800et_nat @ Q8 @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_4041_one__assn__raw_Ocases,axiom,
    ! [X2: produc3658429121746597890et_nat] :
      ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X2
         != ( produc7507926704131184380et_nat @ H3 @ As ) ) ).

% one_assn_raw.cases
thf(fact_4042_subset__minus__empty,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ( minus_minus_set_real @ A2 @ B5 )
        = bot_bot_set_real ) ) ).

% subset_minus_empty
thf(fact_4043_subset__minus__empty,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( minus_minus_set_nat @ A2 @ B5 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_4044_subset__minus__empty,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( minus_minus_set_int @ A2 @ B5 )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_4045_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_4046_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H2 )
      = ( ( produc8586169260539613262et_nat @ H2 )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_4047_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_4048_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4049_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4050_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4051_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4052_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4053_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4054_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4055_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4056_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4057_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4058_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4059_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4060_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4061_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4062_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_4063_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X: complex] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_4064_Collect__mono__iff,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
      = ( ! [X: product_prod_int_int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_4065_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_4066_set__eq__subset,axiom,
    ( ( ^ [Y5: set_int,Z4: set_int] : Y5 = Z4 )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_4067_subset__trans,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_4068_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_4069_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_4070_Collect__mono,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_4071_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_4072_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_4073_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_4074_subset__iff,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
        ! [T2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ T2 @ A6 )
         => ( member_VEBT_VEBT @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_4075_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A6 )
         => ( member_real @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_4076_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A6 )
         => ( member_int @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_4077_Set_OequalityD2,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_int @ B5 @ A2 ) ) ).

% Set.equalityD2
thf(fact_4078_equalityD1,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% equalityD1
thf(fact_4079_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A6 )
         => ( member_nat @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_4080_subset__eq,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
        ! [X: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X @ A6 )
         => ( member_VEBT_VEBT @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_4081_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A6 )
         => ( member_real @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_4082_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A6 )
         => ( member_int @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_4083_equalityE,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( A2 = B5 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
         => ~ ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).

% equalityE
thf(fact_4084_subsetD,axiom,
    ! [A2: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_4085_subsetD,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
     => ( ( member_VEBT_VEBT @ C @ A2 )
       => ( member_VEBT_VEBT @ C @ B5 ) ) ) ).

% subsetD
thf(fact_4086_subsetD,axiom,
    ! [A2: set_real,B5: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B5 ) ) ) ).

% subsetD
thf(fact_4087_subsetD,axiom,
    ! [A2: set_int,B5: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B5 ) ) ) ).

% subsetD
thf(fact_4088_in__mono,axiom,
    ! [A2: set_nat,B5: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_nat @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_4089_in__mono,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
     => ( ( member_VEBT_VEBT @ X2 @ A2 )
       => ( member_VEBT_VEBT @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_4090_in__mono,axiom,
    ! [A2: set_real,B5: set_real,X2: real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ( member_real @ X2 @ A2 )
       => ( member_real @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_4091_in__mono,axiom,
    ! [A2: set_int,B5: set_int,X2: int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( member_int @ X2 @ A2 )
       => ( member_int @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_4092_double__diff,axiom,
    ! [A2: set_nat,B5: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ C4 )
       => ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_4093_double__diff,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_4094_Diff__subset,axiom,
    ! [A2: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ A2 ) ).

% Diff_subset
thf(fact_4095_Diff__subset,axiom,
    ! [A2: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ A2 ) ).

% Diff_subset
thf(fact_4096_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D5 @ B5 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ ( minus_minus_set_nat @ C4 @ D5 ) ) ) ) ).

% Diff_mono
thf(fact_4097_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D5: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ( ord_less_eq_set_int @ D5 @ B5 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ ( minus_minus_set_int @ C4 @ D5 ) ) ) ) ).

% Diff_mono
thf(fact_4098_verit__la__generic,axiom,
    ! [A: int,X2: int] :
      ( ( ord_less_eq_int @ A @ X2 )
      | ( A = X2 )
      | ( ord_less_eq_int @ X2 @ A ) ) ).

% verit_la_generic
thf(fact_4099_psubset__imp__ex__mem,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A2 @ B5 )
     => ? [B2: vEBT_VEBT] : ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4100_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ( ord_less_set_real @ A2 @ B5 )
     => ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4101_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4102_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B5 )
     => ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4103_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_4104_less__eq__set__def,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( ord_le418104280809901481VEBT_o
          @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A6 )
          @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_4105_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_4106_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_4107_Collect__subset,axiom,
    ! [A2: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ord_le4337996190870823476T_VEBT
      @ ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4108_Collect__subset,axiom,
    ! [A2: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4109_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4110_Collect__subset,axiom,
    ! [A2: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X: complex] :
            ( ( member_complex @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4111_Collect__subset,axiom,
    ! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4112_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4113_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A5: real] :
      ( ( ~ ( ord_less_eq_real @ B3 @ A5 ) )
      = ( ord_less_real @ A5 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4114_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A5: rat] :
      ( ( ~ ( ord_less_eq_rat @ B3 @ A5 ) )
      = ( ord_less_rat @ A5 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4115_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A5: num] :
      ( ( ~ ( ord_less_eq_num @ B3 @ A5 ) )
      = ( ord_less_num @ A5 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4116_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B3 @ A5 ) )
      = ( ord_less_nat @ A5 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4117_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A5: int] :
      ( ( ~ ( ord_less_eq_int @ B3 @ A5 ) )
      = ( ord_less_int @ A5 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4118_verit__sum__simplify,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% verit_sum_simplify
thf(fact_4119_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_4120_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_4121_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_4122_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_4123_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] :
      ( one
     != ( bit0 @ X23 ) ) ).

% verit_eq_simplify(10)
thf(fact_4124_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X33: num] :
      ( ( bit0 @ X23 )
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(14)
thf(fact_4125_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] :
      ( one
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(12)
thf(fact_4126_psubsetE,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
         => ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).

% psubsetE
thf(fact_4127_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_4128_psubset__imp__subset,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% psubset_imp_subset
thf(fact_4129_psubset__subset__trans,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_4130_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_4131_subset__psubset__trans,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_4132_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_4133_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_4134_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_o,T: nat,T3: nat] :
      ( ( time_TBOUND_o @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_TBOUND_o @ C @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_4135_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_nat,T: nat,T3: nat] :
      ( ( time_TBOUND_nat @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_TBOUND_nat @ C @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_4136_TBOUND__mono,axiom,
    ! [C: heap_T2636463487746394924on_nat,T: nat,T3: nat] :
      ( ( time_T8353473612707095248on_nat @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_T8353473612707095248on_nat @ C @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_4137_TBOUND__mono,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,T: nat,T3: nat] :
      ( ( time_T5737551269749752165_VEBTi @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_T5737551269749752165_VEBTi @ C @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_4138_htt__htD,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ P @ C @ Q @ T )
     => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_4139_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ P @ C @ Q @ T )
     => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_4140_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ P @ C @ Q @ T )
     => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_4141_htt__htD,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T )
     => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_4142_htt__htD,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn,T: nat] :
      ( ( time_h7375078312994015913t_unit @ P @ C @ Q @ T )
     => ( hoare_8945653483474564448t_unit @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_4143_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4144_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4145_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4146_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4147_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4148_norm__pre__ex__rule__htt,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ! [X3: list_VEBT_VEBTi] : ( time_htt_option_nat @ ( P @ X3 ) @ F @ Q @ T )
     => ( time_htt_option_nat @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q @ T ) ) ).

% norm_pre_ex_rule_htt
thf(fact_4149_norm__pre__ex__rule__htt,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ! [X3: list_VEBT_VEBTi] : ( time_htt_nat @ ( P @ X3 ) @ F @ Q @ T )
     => ( time_htt_nat @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q @ T ) ) ).

% norm_pre_ex_rule_htt
thf(fact_4150_norm__pre__ex__rule__htt,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ! [X3: list_VEBT_VEBTi] : ( time_htt_VEBT_VEBTi @ ( P @ X3 ) @ F @ Q @ T )
     => ( time_htt_VEBT_VEBTi @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q @ T ) ) ).

% norm_pre_ex_rule_htt
thf(fact_4151_norm__pre__ex__rule__htt,axiom,
    ! [P: list_VEBT_VEBTi > assn,F: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ! [X3: list_VEBT_VEBTi] : ( time_htt_o @ ( P @ X3 ) @ F @ Q @ T )
     => ( time_htt_o @ ( ex_ass463751140784270563_VEBTi @ P ) @ F @ Q @ T ) ) ).

% norm_pre_ex_rule_htt
thf(fact_4152_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_T2636463487746394924on_nat,Q: list_VEBT_VEBTi > option_nat > assn,X2: list_VEBT_VEBTi,T: nat] :
      ( ( time_htt_option_nat @ P @ F @ ( Q @ X2 ) @ T )
     => ( time_htt_option_nat @ P @ F
        @ ^ [R: option_nat] :
            ( ex_ass463751140784270563_VEBTi
            @ ^ [X: list_VEBT_VEBTi] : ( Q @ X @ R ) )
        @ T ) ) ).

% norm_post_ex_rule_htt
thf(fact_4153_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,Q: list_VEBT_VEBTi > nat > assn,X2: list_VEBT_VEBTi,T: nat] :
      ( ( time_htt_nat @ P @ F @ ( Q @ X2 ) @ T )
     => ( time_htt_nat @ P @ F
        @ ^ [R: nat] :
            ( ex_ass463751140784270563_VEBTi
            @ ^ [X: list_VEBT_VEBTi] : ( Q @ X @ R ) )
        @ T ) ) ).

% norm_post_ex_rule_htt
thf(fact_4154_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_T8145700208782473153_VEBTi,Q: list_VEBT_VEBTi > vEBT_VEBTi > assn,X2: list_VEBT_VEBTi,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ P @ F @ ( Q @ X2 ) @ T )
     => ( time_htt_VEBT_VEBTi @ P @ F
        @ ^ [R: vEBT_VEBTi] :
            ( ex_ass463751140784270563_VEBTi
            @ ^ [X: list_VEBT_VEBTi] : ( Q @ X @ R ) )
        @ T ) ) ).

% norm_post_ex_rule_htt
thf(fact_4155_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_Time_Heap_o,Q: list_VEBT_VEBTi > $o > assn,X2: list_VEBT_VEBTi,T: nat] :
      ( ( time_htt_o @ P @ F @ ( Q @ X2 ) @ T )
     => ( time_htt_o @ P @ F
        @ ^ [R: $o] :
            ( ex_ass463751140784270563_VEBTi
            @ ^ [X: list_VEBT_VEBTi] : ( Q @ X @ R ) )
        @ T ) ) ).

% norm_post_ex_rule_htt
thf(fact_4156_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% int_ops(3)
thf(fact_4157_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_4158_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_4159_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_4160_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_4161_int__plus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_4162_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_4163_TBOUND__return,axiom,
    ! [X2: option_nat] : ( time_T8353473612707095248on_nat @ ( heap_T3487192422709364219on_nat @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4164_TBOUND__return,axiom,
    ! [X2: nat] : ( time_TBOUND_nat @ ( heap_Time_return_nat @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4165_TBOUND__return,axiom,
    ! [X2: $o] : ( time_TBOUND_o @ ( heap_Time_return_o @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4166_TBOUND__return,axiom,
    ! [X2: vEBT_VEBTi] : ( time_T5737551269749752165_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4167_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_4168_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_4169_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_4170_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_4171_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_4172_TBOUND__nth,axiom,
    ! [Xs2: array_o,I: nat] : ( time_TBOUND_o @ ( array_nth_o @ Xs2 @ I ) @ one_one_nat ) ).

% TBOUND_nth
thf(fact_4173_TBOUND__nth,axiom,
    ! [Xs2: array_nat,I: nat] : ( time_TBOUND_nat @ ( array_nth_nat @ Xs2 @ I ) @ one_one_nat ) ).

% TBOUND_nth
thf(fact_4174_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_4175_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_4176_TBOUNDD,axiom,
    ! [M: heap_T5738788834812785303t_unit,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T7469515765551943773t_unit @ M @ T )
     => ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_4177_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_o @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_o @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_4178_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_4179_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T8353473612707095248on_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_4180_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T5737551269749752165_VEBTi @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_4181_TBOUNDI,axiom,
    ! [M: heap_T5738788834812785303t_unit,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ M @ H3 ) @ T )
     => ( time_T7469515765551943773t_unit @ M @ T ) ) ).

% TBOUNDI
thf(fact_4182_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M @ H3 ) @ T )
     => ( time_TBOUND_o @ M @ T ) ) ).

% TBOUNDI
thf(fact_4183_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M @ H3 ) @ T )
     => ( time_TBOUND_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_4184_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H3 ) @ T )
     => ( time_T8353473612707095248on_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_4185_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H3 ) @ T )
     => ( time_T5737551269749752165_VEBTi @ M @ T ) ) ).

% TBOUNDI
thf(fact_4186_TBOUND__def,axiom,
    ( time_T7469515765551943773t_unit
    = ( ^ [M7: heap_T5738788834812785303t_unit,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ M7 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_4187_TBOUND__def,axiom,
    ( time_TBOUND_o
    = ( ^ [M7: heap_Time_Heap_o,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M7 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_4188_TBOUND__def,axiom,
    ( time_TBOUND_nat
    = ( ^ [M7: heap_Time_Heap_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M7 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_4189_TBOUND__def,axiom,
    ( time_T8353473612707095248on_nat
    = ( ^ [M7: heap_T2636463487746394924on_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M7 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_4190_TBOUND__def,axiom,
    ( time_T5737551269749752165_VEBTi
    = ( ^ [M7: heap_T8145700208782473153_VEBTi,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M7 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_4191_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_option_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_4192_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_4193_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_VEBT_VEBTi @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_4194_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_o @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_4195_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_option_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_4196_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_4197_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_VEBT_VEBTi @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_4198_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_o @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_4199_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_4200_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_4201_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ( time_T8353473612707095248on_nat @ C @ T )
       => ( time_htt_option_nat @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_4202_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ( time_TBOUND_nat @ C @ T )
       => ( time_htt_nat @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_4203_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ( time_TBOUND_o @ C @ T )
       => ( time_htt_o @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_4204_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ( time_T5737551269749752165_VEBTi @ C @ T )
       => ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_4205_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn,T: nat] :
      ( ( hoare_8945653483474564448t_unit @ P @ C @ Q )
     => ( ( time_T7469515765551943773t_unit @ C @ T )
       => ( time_h7375078312994015913t_unit @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_4206_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_o,T: nat] :
      ( ( P
       => ( time_TBOUND_o @ M @ T ) )
     => ( time_TBOUND_o
        @ ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ T ) ) ).

% TBOUND_assert'_weak
thf(fact_4207_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,T: nat] :
      ( ( P
       => ( time_TBOUND_nat @ M @ T ) )
     => ( time_TBOUND_nat
        @ ( heap_T3781436268274291734it_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ T ) ) ).

% TBOUND_assert'_weak
thf(fact_4208_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,T: nat] :
      ( ( P
       => ( time_T8353473612707095248on_nat @ M @ T ) )
     => ( time_T8353473612707095248on_nat
        @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ T ) ) ).

% TBOUND_assert'_weak
thf(fact_4209_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ( P
       => ( time_T5737551269749752165_VEBTi @ M @ T ) )
     => ( time_T5737551269749752165_VEBTi
        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ T ) ) ).

% TBOUND_assert'_weak
thf(fact_4210_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_4211_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_4212_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_o,T: nat] :
      ( ( P
       => ( time_TBOUND_o @ M @ T ) )
     => ( time_TBOUND_o
        @ ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ ( if_nat @ P @ T @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4213_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,T: nat] :
      ( ( P
       => ( time_TBOUND_nat @ M @ T ) )
     => ( time_TBOUND_nat
        @ ( heap_T3781436268274291734it_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ ( if_nat @ P @ T @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4214_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,T: nat] :
      ( ( P
       => ( time_T8353473612707095248on_nat @ M @ T ) )
     => ( time_T8353473612707095248on_nat
        @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ ( if_nat @ P @ T @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4215_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ( P
       => ( time_T5737551269749752165_VEBTi @ M @ T ) )
     => ( time_T5737551269749752165_VEBTi
        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ ( if_nat @ P @ T @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4216_TBOUND__option__case,axiom,
    ! [T: option_nat,F: heap_Time_Heap_o,Bnd: nat,F3: nat > heap_Time_Heap_o,Bnd2: nat > nat] :
      ( ( ( T = none_nat )
       => ( time_TBOUND_o @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T
              = ( some_nat @ X3 ) )
           => ( time_TBOUND_o @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_o @ ( case_o6892868863119666303_o_nat @ F @ F3 @ T ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4217_TBOUND__option__case,axiom,
    ! [T: option_num,F: heap_Time_Heap_o,Bnd: nat,F3: num > heap_Time_Heap_o,Bnd2: num > nat] :
      ( ( ( T = none_num )
       => ( time_TBOUND_o @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T
              = ( some_num @ X3 ) )
           => ( time_TBOUND_o @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_o @ ( case_o3450200649275444937_o_num @ F @ F3 @ T ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4218_TBOUND__option__case,axiom,
    ! [T: option_nat,F: heap_Time_Heap_nat,Bnd: nat,F3: nat > heap_Time_Heap_nat,Bnd2: nat > nat] :
      ( ( ( T = none_nat )
       => ( time_TBOUND_nat @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T
              = ( some_nat @ X3 ) )
           => ( time_TBOUND_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_nat @ ( case_o6609685678014844897at_nat @ F @ F3 @ T ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4219_TBOUND__option__case,axiom,
    ! [T: option_num,F: heap_Time_Heap_nat,Bnd: nat,F3: num > heap_Time_Heap_nat,Bnd2: num > nat] :
      ( ( ( T = none_num )
       => ( time_TBOUND_nat @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T
              = ( some_num @ X3 ) )
           => ( time_TBOUND_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_nat @ ( case_o3167017464170623531at_num @ F @ F3 @ T ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4220_TBOUND__option__case,axiom,
    ! [T: option_nat,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: nat > heap_T8145700208782473153_VEBTi,Bnd2: nat > nat] :
      ( ( ( T = none_nat )
       => ( time_T5737551269749752165_VEBTi @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T
              = ( some_nat @ X3 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( case_o3780387683879180358Ti_nat @ F @ F3 @ T ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4221_TBOUND__option__case,axiom,
    ! [T: option_num,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: num > heap_T8145700208782473153_VEBTi,Bnd2: num > nat] :
      ( ( ( T = none_num )
       => ( time_T5737551269749752165_VEBTi @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T
              = ( some_num @ X3 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( case_o337719470034958992Ti_num @ F @ F3 @ T ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4222_TBOUND__option__case,axiom,
    ! [T: option_nat,F: heap_T2636463487746394924on_nat,Bnd: nat,F3: nat > heap_T2636463487746394924on_nat,Bnd2: nat > nat] :
      ( ( ( T = none_nat )
       => ( time_T8353473612707095248on_nat @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T
              = ( some_nat @ X3 ) )
           => ( time_T8353473612707095248on_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T8353473612707095248on_nat @ ( case_o2256915875499652529at_nat @ F @ F3 @ T ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4223_TBOUND__option__case,axiom,
    ! [T: option_num,F: heap_T2636463487746394924on_nat,Bnd: nat,F3: num > heap_T2636463487746394924on_nat,Bnd2: num > nat] :
      ( ( ( T = none_num )
       => ( time_T8353473612707095248on_nat @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T
              = ( some_num @ X3 ) )
           => ( time_T8353473612707095248on_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T8353473612707095248on_nat @ ( case_o8037619698510206971at_num @ F @ F3 @ T ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4224_TBOUND__option__case,axiom,
    ! [T: option4927543243414619207at_nat,F: heap_Time_Heap_nat,Bnd: nat,F3: product_prod_nat_nat > heap_Time_Heap_nat,Bnd2: product_prod_nat_nat > nat] :
      ( ( ( T = none_P5556105721700978146at_nat )
       => ( time_TBOUND_nat @ F @ Bnd ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( T
              = ( some_P7363390416028606310at_nat @ X3 ) )
           => ( time_TBOUND_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_nat @ ( case_o3959993630158478256at_nat @ F @ F3 @ T ) @ ( case_o2098746482150326116at_nat @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4225_TBOUND__option__case,axiom,
    ! [T: option4927543243414619207at_nat,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: product_prod_nat_nat > heap_T8145700208782473153_VEBTi,Bnd2: product_prod_nat_nat > nat] :
      ( ( ( T = none_P5556105721700978146at_nat )
       => ( time_T5737551269749752165_VEBTi @ F @ Bnd ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( T
              = ( some_P7363390416028606310at_nat @ X3 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( case_o1356590567247012107at_nat @ F @ F3 @ T ) @ ( case_o2098746482150326116at_nat @ Bnd @ Bnd2 @ T ) ) ) ) ).

% TBOUND_option_case
thf(fact_4226_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A3
                       => ( Y2
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y2 = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( ( B2
                           => ( Y2
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B2
                           => ( ( A3
                               => ( Y2
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A3
                               => ( Y2 = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y2 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( 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 @ TreeList2 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_4227_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( 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 ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_4228_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2
                    = ( vEBT_Leaf @ $false @ B2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2
                      = ( vEBT_Leaf @ A3 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2
                          = ( vEBT_Leaf @ A3 @ B2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                     => ( ( Y2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                       => ( ( Y2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( 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 ) ) @ TreeList2 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                                  & ( ~ ( ( 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_4229_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 ) )
       => ( ! [Uu2: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B2 ) @ 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 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ 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 @ Vc @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( 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 @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_4230_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y2
                      = ( vEBT_Leaf @ $true @ B2 ) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y2
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y2
                          = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
               => ( ( Y2
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
                 => ( ( Y2
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 ) )
                              & ~ ( ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_4231_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( 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 @ TreeList2 ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_4232_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2 = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_4233_Diff__idemp,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ B5 )
      = ( minus_minus_set_nat @ A2 @ B5 ) ) ).

% Diff_idemp
thf(fact_4234_Diff__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
      = ( ( member_VEBT_VEBT @ C @ A2 )
        & ~ ( member_VEBT_VEBT @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_4235_Diff__iff,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
      = ( ( member_real @ C @ A2 )
        & ~ ( member_real @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_4236_Diff__iff,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_4237_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_4238_DiffI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ A2 )
     => ( ~ ( member_VEBT_VEBT @ C @ B5 )
       => ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4239_DiffI,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ A2 )
     => ( ~ ( member_real @ C @ B5 )
       => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4240_DiffI,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B5 )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4241_DiffI,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B5 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4242_minus__set__def,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( minus_2794559001203777698VEBT_o
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A6 )
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4243_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X: real] : ( member_real @ X @ A6 )
            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4244_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X: int] : ( member_int @ X @ A6 )
            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4245_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X: complex] : ( member_complex @ X @ A6 )
            @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4246_minus__set__def,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( minus_711738161318947805_int_o
            @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A6 )
            @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4247_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A6 )
            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4248_set__diff__eq,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ A6 )
              & ~ ( member_VEBT_VEBT @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4249_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A6 )
              & ~ ( member_real @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4250_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A6 )
              & ~ ( member_int @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4251_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ A6 )
              & ~ ( member_complex @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4252_set__diff__eq,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A6 )
              & ~ ( member5262025264175285858nt_int @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4253_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A6 )
              & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4254_DiffD2,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
     => ~ ( member_VEBT_VEBT @ C @ B5 ) ) ).

% DiffD2
thf(fact_4255_DiffD2,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
     => ~ ( member_real @ C @ B5 ) ) ).

% DiffD2
thf(fact_4256_DiffD2,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
     => ~ ( member_int @ C @ B5 ) ) ).

% DiffD2
thf(fact_4257_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
     => ~ ( member_nat @ C @ B5 ) ) ).

% DiffD2
thf(fact_4258_DiffD1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
     => ( member_VEBT_VEBT @ C @ A2 ) ) ).

% DiffD1
thf(fact_4259_DiffD1,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
     => ( member_real @ C @ A2 ) ) ).

% DiffD1
thf(fact_4260_DiffD1,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_4261_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_4262_DiffE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
     => ~ ( ( member_VEBT_VEBT @ C @ A2 )
         => ( member_VEBT_VEBT @ C @ B5 ) ) ) ).

% DiffE
thf(fact_4263_DiffE,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
     => ~ ( ( member_real @ C @ A2 )
         => ( member_real @ C @ B5 ) ) ) ).

% DiffE
thf(fact_4264_DiffE,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B5 ) ) ) ).

% DiffE
thf(fact_4265_DiffE,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B5 ) ) ) ).

% DiffE
thf(fact_4266_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( 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 @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
                   => ( ( 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 @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_4267_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2 = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_4268_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2
                  = ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y2
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
                   => ( ~ Y2
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_4269_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A3 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B2 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) @ Xa ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ 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 @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_4270_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2
                  = ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
                 => ( ( Y2
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_4271_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_4272_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A3 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B2 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_4273_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ 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 @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_4274_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ 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 @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_4275_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y2
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y2
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ 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 @ Va2 @ Vb2 ) @ Xa ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) )
                   => ( ( 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 @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ Xa ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( Y2
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_4276_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,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ 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 @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_4277_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_4278_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_4279_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_4280_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_4281_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_4282_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_4283_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4284_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4285_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4286_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4287_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4288_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4289_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4290_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4291_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4292_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4293_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4294_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4295_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4296_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4297_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4298_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4299_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4300_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4301_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4302_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4303_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4304_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4305_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4306_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4307_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4308_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4309_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4310_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4311_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4312_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4313_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4314_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4315_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4316_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4317_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_4318_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_4319_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_4320_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_4321_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_4322_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_4323_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_4324_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_4325_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_4326_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_4327_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_4328_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_4329_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_4330_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_4331_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_4332_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_4333_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( A4
          = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4334_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( A4
          = ( ord_max_num @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4335_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4336_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( A4
          = ( ord_max_int @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4337_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4338_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4339_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4340_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4341_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4342_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4343_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4344_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4345_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_4346_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_4347_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_4348_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_4349_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_4350_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_4351_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_4352_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_4353_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4354_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4355_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4356_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4357_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_4358_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_4359_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_4360_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_4361_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_4362_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4363_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4364_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4365_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4366_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4367_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( A4
            = ( ord_max_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4368_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( A4
            = ( ord_max_rat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4369_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( A4
            = ( ord_max_num @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4370_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( ord_max_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4371_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( A4
            = ( ord_max_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4372_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4373_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4374_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4375_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4376_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4377_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4378_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4379_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4380_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4381_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4382_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_4383_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_4384_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_4385_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_4386_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_4387_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_4388_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_4389_set__bit__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( bit_se4894374433684937756l_num1 @ zero_zero_nat @ A )
      = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4390_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4391_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4392_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( ( Phi
         => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) )
       => ( hoare_7629718768684598413on_nat @ P
          @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ Phi )
            @ ^ [Uu3: product_unit] : C )
          @ Q ) ) ) ).

% assert'_bind_rule
thf(fact_4393_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_Time_Heap_nat,Q: nat > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( ( Phi
         => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) )
       => ( hoare_3067605981109127869le_nat @ P
          @ ( heap_T3781436268274291734it_nat @ ( refine_Imp_assert @ Phi )
            @ ^ [Uu3: product_unit] : C )
          @ Q ) ) ) ).

% assert'_bind_rule
thf(fact_4394_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( ( Phi
         => ( hoare_hoare_triple_o @ P @ C @ Q ) )
       => ( hoare_hoare_triple_o @ P
          @ ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ Phi )
            @ ^ [Uu3: product_unit] : C )
          @ Q ) ) ) ).

% assert'_bind_rule
thf(fact_4395_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( ( Phi
         => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) )
       => ( hoare_1429296392585015714_VEBTi @ P
          @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ Phi )
            @ ^ [Uu3: product_unit] : C )
          @ Q ) ) ) ).

% assert'_bind_rule
thf(fact_4396_assert_H__bind__rule,axiom,
    ! [P: assn,Phi: $o,C: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( ( Phi
         => ( hoare_8945653483474564448t_unit @ P @ C @ Q ) )
       => ( hoare_8945653483474564448t_unit @ P
          @ ( heap_T2633723481742716231t_unit @ ( refine_Imp_assert @ Phi )
            @ ^ [Uu3: product_unit] : C )
          @ Q ) ) ) ).

% assert'_bind_rule
thf(fact_4397_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c @ T @ 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_4398_div__of__0__id,axiom,
    ! [N2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
      = zero_z3563351764282998399l_num1 ) ).

% div_of_0_id
thf(fact_4399_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_4400_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_4401_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_4402_word__less__1,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ one_on7727431528512463931l_num1 )
      = ( X2 = zero_z3563351764282998399l_num1 ) ) ).

% word_less_1
thf(fact_4403_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4404_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4405_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_4406_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_4407_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_4408_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_4409_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_4410_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_4411_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_4412_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_4413_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_4414_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_4415_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_4416_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_4417_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_4418_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_4419_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_4420_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_4421_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_4422_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_4423_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_4424_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_4425_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_4426_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_4427_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_4428_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_4429_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_4430_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_4431_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_4432_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_4433_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_4434_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_4435_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_4436_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_4437_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_4438_word__subset__less,axiom,
    ! [X2: word_N3645301735248828278l_num1,R3: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
      ( ( ord_le5203802739334966412l_num1 @ ( set_or6221694504095523457l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ R3 ) @ one_on7727431528512463931l_num1 ) ) @ ( set_or6221694504095523457l_num1 @ Y2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ Y2 @ S3 ) @ one_on7727431528512463931l_num1 ) ) )
     => ( ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ R3 ) @ one_on7727431528512463931l_num1 ) )
       => ( ( ord_le3335648743751981014l_num1 @ Y2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ Y2 @ S3 ) @ one_on7727431528512463931l_num1 ) )
         => ( ( S3 != zero_z3563351764282998399l_num1 )
           => ( ord_le3335648743751981014l_num1 @ R3 @ S3 ) ) ) ) ) ).

% word_subset_less
thf(fact_4439_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_4440_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_4441_word__less__cases,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ Y2 )
     => ( ( X2
          = ( minus_4019991460397169231l_num1 @ Y2 @ one_on7727431528512463931l_num1 ) )
        | ( ord_le750835935415966154l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Y2 @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_less_cases
thf(fact_4442_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_4443_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_4444_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_4445_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_4446_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_4447_word__le__minus__one__leq,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ Y2 )
     => ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Y2 @ one_on7727431528512463931l_num1 ) ) ) ).

% word_le_minus_one_leq
thf(fact_4448_word__minus__one__le__leq,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) @ Y2 )
     => ( ord_le3335648743751981014l_num1 @ X2 @ Y2 ) ) ).

% word_minus_one_le_leq
thf(fact_4449_le__step__down__word,axiom,
    ! [I: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ I @ N2 )
     => ( ( I != N2 )
       => ( ord_le3335648743751981014l_num1 @ I @ ( minus_4019991460397169231l_num1 @ N2 @ one_on7727431528512463931l_num1 ) ) ) ) ).

% le_step_down_word
thf(fact_4450_le__step__down__word__2,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Y2 @ one_on7727431528512463931l_num1 ) ) ) ) ).

% le_step_down_word_2
thf(fact_4451_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_4452_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_4453_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_4454_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_4455_word__sub__plus__one__nonzero,axiom,
    ! [N5: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ N5 @ N2 )
     => ( ( N5 != zero_z3563351764282998399l_num1 )
       => ( ( plus_p361126936061061375l_num1 @ ( minus_4019991460397169231l_num1 @ N2 @ N5 ) @ one_on7727431528512463931l_num1 )
         != zero_z3563351764282998399l_num1 ) ) ) ).

% word_sub_plus_one_nonzero
thf(fact_4456_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_4457_More__Word_Oword__div__mult,axiom,
    ! [C: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ C )
     => ( ( ord_le750835935415966154l_num1 @ A @ ( times_7065122842183080059l_num1 @ B @ C ) )
       => ( ord_le750835935415966154l_num1 @ ( divide1791077408188789448l_num1 @ A @ C ) @ B ) ) ) ).

% More_Word.word_div_mult
thf(fact_4458_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_4459_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_4460_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_4461_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_4462_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_4463_less__is__non__zero__p1,axiom,
    ! [A: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ A @ K )
     => ( ( plus_p361126936061061375l_num1 @ A @ one_on7727431528512463931l_num1 )
       != zero_z3563351764282998399l_num1 ) ) ).

% less_is_non_zero_p1
thf(fact_4464_word__div__1,axiom,
    ! [N2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ N2 @ one_on7727431528512463931l_num1 )
      = N2 ) ).

% word_div_1
thf(fact_4465_div__by__0__word,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ X2 @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% div_by_0_word
thf(fact_4466_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_4467_lt1__neq0,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ X2 )
      = ( X2 != zero_z3563351764282998399l_num1 ) ) ).

% lt1_neq0
thf(fact_4468_div__to__mult__word__lt,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ ( divide1791077408188789448l_num1 @ Y2 @ Z ) )
     => ( ord_le3335648743751981014l_num1 @ ( times_7065122842183080059l_num1 @ X2 @ Z ) @ Y2 ) ) ).

% div_to_mult_word_lt
thf(fact_4469_word__le__plus__1,axiom,
    ! [Y2: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Y2 @ ( plus_p361126936061061375l_num1 @ Y2 @ N2 ) )
     => ( ( ord_le750835935415966154l_num1 @ A @ N2 )
       => ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ Y2 @ A ) @ ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ Y2 @ A ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_le_plus_1
thf(fact_4470_plus__one__helper,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ ( plus_p361126936061061375l_num1 @ N2 @ one_on7727431528512463931l_num1 ) )
     => ( ord_le3335648743751981014l_num1 @ X2 @ N2 ) ) ).

% plus_one_helper
thf(fact_4471_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_4472_word__1__0,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,X2: nat] :
      ( ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ A @ one_on7727431528512463931l_num1 ) @ B )
     => ( ( ord_le750835935415966154l_num1 @ A @ ( semiri8819519690708144855l_num1 @ X2 ) )
       => ( ord_le750835935415966154l_num1 @ A @ B ) ) ) ).

% word_1_0
thf(fact_4473_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_4474_inc__le,axiom,
    ! [I: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ I @ M )
     => ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ I @ one_on7727431528512463931l_num1 ) @ M ) ) ).

% inc_le
thf(fact_4475_inc__i,axiom,
    ! [I: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ I )
     => ( ( ord_le750835935415966154l_num1 @ I @ M )
       => ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ ( plus_p361126936061061375l_num1 @ I @ one_on7727431528512463931l_num1 ) )
          & ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ I @ one_on7727431528512463931l_num1 ) @ M ) ) ) ) ).

% inc_i
thf(fact_4476_word__div__mult__le,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ ( times_7065122842183080059l_num1 @ ( divide1791077408188789448l_num1 @ A @ B ) @ B ) @ A ) ).

% word_div_mult_le
thf(fact_4477_i0__lb,axiom,
    ! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).

% i0_lb
thf(fact_4478_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_4479_div__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
     => ( ( divide1791077408188789448l_num1 @ W @ W )
        = one_on7727431528512463931l_num1 ) ) ).

% div_word_self
thf(fact_4480_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_4481_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_4482_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_4483_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_4484_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_4485_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_4486_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_4487_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_4488_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_4489_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_4490_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_4491_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_4492_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_4493_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_4494_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_4495_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_4496_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu @ B ) @ 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_4497_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,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_4498_Abs__fnat__hom__1,axiom,
    ( one_on7727431528512463931l_num1
    = ( semiri8819519690708144855l_num1 @ ( suc @ zero_zero_nat ) ) ) ).

% Abs_fnat_hom_1
thf(fact_4499_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_4500_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( hoare_8945653483474564448t_unit @ P @ ( refine_Imp_assert @ Phi )
        @ ^ [Uu3: product_unit] : P ) ) ).

% assert'_rule
thf(fact_4501_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_rat @ ( semiri681578069525770553at_rat @ A ) @ ( semiri681578069525770553at_rat @ B ) )
      = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4502_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
      = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4503_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
      = ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4504_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ A ) @ ( semiri1316708129612266289at_nat @ B ) )
      = ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4505_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ A ) @ ( semiri4939895301339042750nteger @ B ) )
      = ( semiri4939895301339042750nteger @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4506_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_4507_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_4508_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_4509_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri2565882477558803405uint32 @ K )
       != zero_zero_uint32 )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4510_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_4511_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_4512_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_4513_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_4514_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri4939895301339042750nteger @ K )
       != zero_z3403309356797280102nteger )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4515_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% succ_bound_height
thf(fact_4516_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X2 )
        @ ^ [R: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_succ @ T @ 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_4517_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
        @ ^ [R: $o] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_member @ T @ 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_4518_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ 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_4519_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 )
     => ( ( ? [Uu2: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ B2 ) )
         => ( ( 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 ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( 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 @ TreeList2 ) )
                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_succ @ Summary2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( 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_4520_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_4521_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,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                  @ ( if_nat
                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                       != none_nat )
                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                      @ ( if_nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_4522_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ 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 @ T @ 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_4523_idiff__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_4524_idiff__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = N2 ) ).

% idiff_0_right
thf(fact_4525_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_4526_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_4527_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_4528_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ K ) )
      = ( numera7442385471795722001l_num1 @ K ) ) ).

% of_int_numeral
thf(fact_4529_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_4530_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_4531_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_4532_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_4533_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_4534_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_4535_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_4536_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_4537_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_4538_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_4539_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_4540_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_4541_of__int__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ one_one_int )
    = one_on7727431528512463931l_num1 ) ).

% of_int_1
thf(fact_4542_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_4543_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_4544_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_4545_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_4546_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_4547_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_4548_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_mult
thf(fact_4549_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_4550_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_4551_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_4552_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_4553_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_diff
thf(fact_4554_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_4555_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X2 )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X2
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4556_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X2 )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X2
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4557_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X2 )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X2
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4558_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X2 ) )
      = ( ( power_power_int @ B @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4559_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X2 ) )
      = ( ( power_power_int @ B @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4560_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X2 ) )
      = ( ( power_power_int @ B @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4561_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_4562_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_uint32 @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_uint32 @ ( ring_1_of_int_uint32 @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4563_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_4564_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_4565_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_4566_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_4567_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_4568_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_4569_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_4570_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_4571_ceiling__add__of__int,axiom,
    ! [X2: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_4572_ceiling__add__of__int,axiom,
    ! [X2: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_4573_ceiling__diff__of__int,axiom,
    ! [X2: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_4574_ceiling__diff__of__int,axiom,
    ! [X2: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_4575_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_4576_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_4577_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_4578_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_4579_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_4580_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_4581_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_4582_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_4583_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_4584_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_4585_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_4586_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_4587_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_4588_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_4589_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_4590_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_4591_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_4592_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_4593_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_4594_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_4595_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_4596_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_4597_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_4598_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_4599_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_4600_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_4601_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_4602_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_4603_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_4604_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_4605_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_4606_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_4607_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_4608_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_4609_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_4610_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_4611_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_4612_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_4613_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_4614_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_4615_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_4616_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_4617_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4618_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4619_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4620_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4621_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4622_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4623_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4624_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4625_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4626_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4627_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4628_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4629_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_4630_diff__nat__numeral,axiom,
    ! [V: num,V3: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).

% diff_nat_numeral
thf(fact_4631_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
        = ( ring_1_of_int_rat @ Z ) ) ) ).

% of_nat_nat
thf(fact_4632_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
        = ( ring_1_of_int_real @ Z ) ) ) ).

% of_nat_nat
thf(fact_4633_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = ( ring_1_of_int_int @ Z ) ) ) ).

% of_nat_nat
thf(fact_4634_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri4939895301339042750nteger @ ( nat2 @ Z ) )
        = ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_nat_nat
thf(fact_4635_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_4636_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_4637_nat__ceiling__le__eq,axiom,
    ! [X2: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A )
      = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_4638_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_4639_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4640_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4641_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4642_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4643_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4644_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4645_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4646_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4647_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4648_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4649_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4650_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4651_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_4652_numeral__power__less__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_4653_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_4654_numeral__power__le__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_4655_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_4656_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_4657_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_4658_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_4659_ex__le__of__int,axiom,
    ! [X2: real] :
    ? [Z3: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_le_of_int
thf(fact_4660_ex__le__of__int,axiom,
    ! [X2: rat] :
    ? [Z3: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_le_of_int
thf(fact_4661_ex__of__int__less,axiom,
    ! [X2: real] :
    ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X2 ) ).

% ex_of_int_less
thf(fact_4662_ex__of__int__less,axiom,
    ! [X2: rat] :
    ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X2 ) ).

% ex_of_int_less
thf(fact_4663_ex__less__of__int,axiom,
    ! [X2: real] :
    ? [Z3: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_less_of_int
thf(fact_4664_ex__less__of__int,axiom,
    ! [X2: rat] :
    ? [Z3: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_less_of_int
thf(fact_4665_mult__of__int__commute,axiom,
    ! [X2: int,Y2: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ Y2 )
      = ( times_times_real @ Y2 @ ( ring_1_of_int_real @ X2 ) ) ) ).

% mult_of_int_commute
thf(fact_4666_mult__of__int__commute,axiom,
    ! [X2: int,Y2: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X2 ) @ Y2 )
      = ( times_times_rat @ Y2 @ ( ring_1_of_int_rat @ X2 ) ) ) ).

% mult_of_int_commute
thf(fact_4667_mult__of__int__commute,axiom,
    ! [X2: int,Y2: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X2 ) @ Y2 )
      = ( times_times_int @ Y2 @ ( ring_1_of_int_int @ X2 ) ) ) ).

% mult_of_int_commute
thf(fact_4668_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_4669_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_4670_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_4671_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z6 ) )
          = ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_4672_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_4673_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
          & ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_4674_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_4675_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_4676_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_4677_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_4678_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_4679_of__nat__ceiling,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_4680_of__nat__ceiling,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ R3 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_4681_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_4682_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_4683_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_4684_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_4685_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_4686_ceiling__le,axiom,
    ! [X2: real,A: int] :
      ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ A ) ) ).

% ceiling_le
thf(fact_4687_ceiling__le,axiom,
    ! [X2: rat,A: int] :
      ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ A ) ) ).

% ceiling_le
thf(fact_4688_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_4689_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_4690_int__minus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_4691_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_4692_real__nat__ceiling__ge,axiom,
    ! [X2: real] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_4693_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_4694_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_4695_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_4696_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_4697_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_4698_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_4699_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_4700_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_4701_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_4702_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_4703_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_4704_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_4705_floor__exists,axiom,
    ! [X2: real] :
    ? [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4706_floor__exists,axiom,
    ! [X2: rat] :
    ? [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4707_of__int__ceiling__le__add__one,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R3 ) ) @ ( plus_plus_real @ R3 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_4708_of__int__ceiling__le__add__one,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R3 ) ) @ ( plus_plus_rat @ R3 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_4709_of__int__ceiling__diff__one__le,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R3 ) ) @ one_one_real ) @ R3 ) ).

% of_int_ceiling_diff_one_le
thf(fact_4710_of__int__ceiling__diff__one__le,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R3 ) ) @ one_one_rat ) @ R3 ) ).

% of_int_ceiling_diff_one_le
thf(fact_4711_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_4712_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_4713_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_4714_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_4715_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_4716_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_4717_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_4718_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_4719_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_4720_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_4721_Suc__as__int,axiom,
    ( suc
    = ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_4722_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_4723_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_4724_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_4725_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_4726_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_4727_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_4728_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N: int,M7: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M7 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_4729_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N: int,M7: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M7 ) ) ) ) ).

% int_less_real_le
thf(fact_4730_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_4731_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_4732_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_4733_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_4734_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_4735_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_4736_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_4737_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_4738_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_4739_ceiling__eq__iff,axiom,
    ! [X2: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X2 )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X2 )
        & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_4740_ceiling__eq__iff,axiom,
    ! [X2: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X2 )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X2 )
        & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_4741_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_4742_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_4743_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_4744_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_4745_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_4746_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_4747_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_4748_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_4749_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_4750_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_4751_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_4752_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% maxt_bound
thf(fact_4753_ceiling__divide__upper,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_4754_ceiling__divide__upper,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_4755_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_4756_mint__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% mint_bound
thf(fact_4757_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_4758_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_4759_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_4760_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ 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_4761_ceiling__divide__lower,axiom,
    ! [Q2: real,P4: 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 @ P4 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_4762_ceiling__divide__lower,axiom,
    ! [Q2: rat,P4: 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 @ P4 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_4763_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_4764_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_4765_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_4766_insersimp,axiom,
    ! [T: vEBT_VEBT,N2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_4767_insertsimp,axiom,
    ! [T: vEBT_VEBT,N2: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insertsimp
thf(fact_4768_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 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y2
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2 != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_4769_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 )
     => ( ! [A3: $o] :
            ( ? [B2: $o] :
                ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y2
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2 != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_4770_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% insert_bound_height
thf(fact_4771_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 ) )
       => ( ! [Uu2: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B2 ) )
             => ( ( 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 @ Uu2 @ B2 ) @ 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 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ 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 @ Vc @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( 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 @ TreeList2 ) )
                                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_succ @ Summary2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_4772_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 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A3: $o,B2: $o] :
                ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ? [N4: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N4 ) ) )
               => ( Y2 != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y2 != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y2 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( 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 @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                          @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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_4773_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,Va: nat,TreeList: list_VEBT_VEBT,Summary: 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 @ Va ) ) @ TreeList @ Summary ) @ 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 @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                    = Ma ) )
                                & ( ( X2 != Mi )
                                 => ( X2 = Ma ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                  = 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 @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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_4774_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 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ A3 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2
               != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ( Y2
                   != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( 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 @ 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 @ 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( 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_4775_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,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                @ ( if_nat
                  @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                     != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                    @ ( if_nat
                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_4776_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 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( 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 ) ) ) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
           => ( Y2 != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y2
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( 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 @ TreeList2 ) )
                            & ~ ( ( Xa = Mi2 )
                                | ( Xa = Ma2 ) ) )
                          @ ( 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 @ 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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_4777_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( 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 ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( 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 @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                              @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa != Mi2 )
                                                         => ( Xa = Ma2 ) ) )
                                                      @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                          = 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 @ Summary2 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_4778_word__of__int__numeral,axiom,
    ! [Bin: num] :
      ( ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ Bin ) )
      = ( numera7442385471795722001l_num1 @ Bin ) ) ).

% word_of_int_numeral
thf(fact_4779_word__of__int__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ one_one_int )
    = one_on7727431528512463931l_num1 ) ).

% word_of_int_1
thf(fact_4780_word__numeral__alt,axiom,
    ( numera7442385471795722001l_num1
    = ( ^ [B4: num] : ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ B4 ) ) ) ) ).

% word_numeral_alt
thf(fact_4781_word__of__int__power__hom,axiom,
    ! [A: int,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ ( ring_17408606157368542149l_num1 @ A ) @ N2 )
      = ( ring_17408606157368542149l_num1 @ ( power_power_int @ A @ N2 ) ) ) ).

% word_of_int_power_hom
thf(fact_4782_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_N_u_l_l @ T ) @ one_one_nat ) ).

% minNull_bound
thf(fact_4783_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_4784_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_4785_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_4786_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_4787_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_4788_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_4789_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_4790_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ 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_4791_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_4792_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_4793_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ Uw ) @ ( 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_4794_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_4795_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 ) )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ( Y2 != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => ( Y2 != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_4796_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% pred_bound_height
thf(fact_4797_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d @ T @ 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_4798_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ~ ( ( X2 = Mi )
                | ( X2 = Ma ) ) )
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
          @ one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_4799_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ 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 @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y2
                          = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ 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 @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( 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 @ 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 @ 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_pred @ Summary2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_4800_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( 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 @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( 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 @ TreeList2 ) )
                                & ~ ( ( Xa = Mi2 )
                                    | ( Xa = Ma2 ) ) )
                              @ ( 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 @ 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ Summary2 @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_4801_delete__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_4802_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K6: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K6 )
         => ( 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 @ K6 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4803_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K6: real,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K6 )
         => ( 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 @ K6 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4804_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_4805_foldr__zero,axiom,
    ! [Xs2: list_nat,D: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
         => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs2 @ I3 ) ) )
     => ( 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_4806_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_4807_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_4808_mod__mod__trivial,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_4809_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_4810_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_4811_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_4812_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_4813_foldr__mono,axiom,
    ! [Xs2: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( ord_less_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
       => ( ( 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_4814_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_4815_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_4816_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_4817_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_4818_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_4819_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_4820_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_4821_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_4822_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_4823_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_4824_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_4825_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_4826_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_4827_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_4828_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_4829_minus__mod__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mod_self2
thf(fact_4830_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_4831_insert__subset,axiom,
    ! [X2: nat,A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B5 )
      = ( ( member_nat @ X2 @ B5 )
        & ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4832_insert__subset,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X2 @ A2 ) @ B5 )
      = ( ( member_VEBT_VEBT @ X2 @ B5 )
        & ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4833_insert__subset,axiom,
    ! [X2: real,A2: set_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A2 ) @ B5 )
      = ( ( member_real @ X2 @ B5 )
        & ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4834_insert__subset,axiom,
    ! [X2: int,A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X2 @ A2 ) @ B5 )
      = ( ( member_int @ X2 @ B5 )
        & ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4835_insert__Diff1,axiom,
    ! [X2: vEBT_VEBT,B5: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ B5 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X2 @ A2 ) @ B5 )
        = ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4836_insert__Diff1,axiom,
    ! [X2: real,B5: set_real,A2: set_real] :
      ( ( member_real @ X2 @ B5 )
     => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ B5 )
        = ( minus_minus_set_real @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4837_insert__Diff1,axiom,
    ! [X2: int,B5: set_int,A2: set_int] :
      ( ( member_int @ X2 @ B5 )
     => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A2 ) @ B5 )
        = ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4838_insert__Diff1,axiom,
    ! [X2: nat,B5: set_nat,A2: set_nat] :
      ( ( member_nat @ X2 @ B5 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B5 )
        = ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4839_Diff__insert0,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
     => ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ B5 ) )
        = ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4840_Diff__insert0,axiom,
    ! [X2: real,A2: set_real,B5: set_real] :
      ( ~ ( member_real @ X2 @ A2 )
     => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ B5 ) )
        = ( minus_minus_set_real @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4841_Diff__insert0,axiom,
    ! [X2: int,A2: set_int,B5: set_int] :
      ( ~ ( member_int @ X2 @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ B5 ) )
        = ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4842_Diff__insert0,axiom,
    ! [X2: nat,A2: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B5 ) )
        = ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4843_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_4844_nat__mod__eq_H,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ A @ N2 )
     => ( ( modulo_modulo_nat @ A @ N2 )
        = A ) ) ).

% nat_mod_eq'
thf(fact_4845_singleton__conv,axiom,
    ! [A: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] : X = A )
      = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv
thf(fact_4846_singleton__conv,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ^ [X: complex] : X = A )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_4847_singleton__conv,axiom,
    ! [A: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] : X = A )
      = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv
thf(fact_4848_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : X = A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_4849_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X: int] : X = A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_4850_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X: real] : X = A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_4851_singleton__conv2,axiom,
    ! [A: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ( ^ [Y5: vEBT_VEBT,Z4: vEBT_VEBT] : Y5 = Z4
          @ A ) )
      = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv2
thf(fact_4852_singleton__conv2,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ( ^ [Y5: complex,Z4: complex] : Y5 = Z4
          @ A ) )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv2
thf(fact_4853_singleton__conv2,axiom,
    ! [A: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ( ^ [Y5: product_prod_int_int,Z4: product_prod_int_int] : Y5 = Z4
          @ A ) )
      = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv2
thf(fact_4854_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z4: nat] : Y5 = Z4
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_4855_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y5: int,Z4: int] : Y5 = Z4
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_4856_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y5: real,Z4: real] : Y5 = Z4
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_4857_bits__mod__by__1,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ A @ one_on7727431528512463931l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_by_1
thf(fact_4858_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_4859_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_4860_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_4861_bits__mod__by__1,axiom,
    ! [A: uint32] :
      ( ( modulo_modulo_uint32 @ A @ one_one_uint32 )
      = zero_zero_uint32 ) ).

% bits_mod_by_1
thf(fact_4862_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_4863_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_4864_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_4865_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_4866_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_4867_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_4868_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_4869_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_4870_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_4871_bits__mod__div__trivial,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ ( modulo1504961113040953224l_num1 @ A @ B ) @ B )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_div_trivial
thf(fact_4872_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_4873_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_4874_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_4875_bits__mod__div__trivial,axiom,
    ! [A: uint32,B: uint32] :
      ( ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A @ B ) @ B )
      = zero_zero_uint32 ) ).

% bits_mod_div_trivial
thf(fact_4876_mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_4877_mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_4878_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_4879_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4880_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4881_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4882_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4883_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4884_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4885_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4886_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4887_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4888_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4889_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4890_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4891_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT )
        = ( insert_VEBT_VEBT @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4892_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4893_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4894_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4895_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A @ A2 )
        = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4896_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4897_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( ( insert_real @ A @ A2 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4898_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4899_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_4900_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( insert_VEBT_VEBT @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4901_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4902_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
      = ( insert_real @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4903_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4904_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_4905_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_4906_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_4907_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_4908_foldr__length,axiom,
    ! [L: list_real] :
      ( ( foldr_real_nat
        @ ^ [X: real] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_real @ L ) ) ).

% foldr_length
thf(fact_4909_foldr__length,axiom,
    ! [L: list_o] :
      ( ( foldr_o_nat
        @ ^ [X: $o] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_o @ L ) ) ).

% foldr_length
thf(fact_4910_foldr__length,axiom,
    ! [L: list_nat] :
      ( ( foldr_nat_nat
        @ ^ [X: nat] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_nat @ L ) ) ).

% foldr_length
thf(fact_4911_foldr__length,axiom,
    ! [L: list_int] :
      ( ( foldr_int_nat
        @ ^ [X: int] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_int @ L ) ) ).

% foldr_length
thf(fact_4912_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_4913_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_4914_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_4915_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_4916_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_4917_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_4918_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_4919_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_4920_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_4921_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_4922_not__mod__2__eq__1__eq__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
       != one_on7727431528512463931l_num1 )
      = ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4923_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4924_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4925_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4926_not__mod__2__eq__1__eq__0,axiom,
    ! [A: uint32] :
      ( ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != one_one_uint32 )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4927_not__mod__2__eq__0__eq__1,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
       != zero_z3563351764282998399l_num1 )
      = ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = one_on7727431528512463931l_num1 ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4928_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4929_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4930_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4931_not__mod__2__eq__0__eq__1,axiom,
    ! [A: uint32] :
      ( ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != zero_zero_uint32 )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4932_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_4933_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_4934_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_4935_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_4936_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_4937_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_4938_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_4939_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_mod
thf(fact_4940_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( insert_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P ) )
      = ( collect_VEBT_VEBT
        @ ^ [U2: vEBT_VEBT] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4941_insert__Collect,axiom,
    ! [A: real,P: real > $o] :
      ( ( insert_real @ A @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4942_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4943_insert__Collect,axiom,
    ! [A: int,P: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4944_insert__Collect,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( insert_complex @ A @ ( collect_complex @ P ) )
      = ( collect_complex
        @ ^ [U2: complex] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4945_insert__Collect,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( insert5033312907999012233nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( collec213857154873943460nt_int
        @ ^ [U2: product_prod_int_int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4946_insert__compr,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A4: vEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( X = A4 )
              | ( member_VEBT_VEBT @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_4947_insert__compr,axiom,
    ( insert_real
    = ( ^ [A4: real,B6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( X = A4 )
              | ( member_real @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_4948_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A4: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A4 )
              | ( member_nat @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_4949_insert__compr,axiom,
    ( insert_int
    = ( ^ [A4: int,B6: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( X = A4 )
              | ( member_int @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_4950_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A4: complex,B6: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( X = A4 )
              | ( member_complex @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_4951_insert__compr,axiom,
    ( insert5033312907999012233nt_int
    = ( ^ [A4: product_prod_int_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( X = A4 )
              | ( member5262025264175285858nt_int @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_4952_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4953_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4954_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4955_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4956_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4957_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4958_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B3 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4959_mod__add__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4960_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4961_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4962_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4963_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4964_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4965_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4966_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4967_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4968_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4969_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4970_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4971_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4972_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4973_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4974_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4975_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4976_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B3 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4977_mod__mult__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4978_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4979_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4980_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4981_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4982_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4983_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4984_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4985_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4986_mod__diff__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4987_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B3 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4988_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_4989_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_4990_power__mod,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N2 ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_4991_power__mod,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N2 ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_4992_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N2 ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_4993_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_4994_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_4995_nat__mod__eq,axiom,
    ! [B: nat,N2: nat,A: nat] :
      ( ( ord_less_nat @ B @ N2 )
     => ( ( ( modulo_modulo_nat @ A @ N2 )
          = ( modulo_modulo_nat @ B @ N2 ) )
       => ( ( modulo_modulo_nat @ A @ N2 )
          = B ) ) ) ).

% nat_mod_eq
thf(fact_4996_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_4997_mod__plus__right,axiom,
    ! [A: nat,X2: nat,M: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ X2 ) @ M )
        = ( modulo_modulo_nat @ ( plus_plus_nat @ B @ X2 ) @ M ) )
      = ( ( modulo_modulo_nat @ A @ M )
        = ( modulo_modulo_nat @ B @ M ) ) ) ).

% mod_plus_right
thf(fact_4998_insert__mono,axiom,
    ! [C4: set_nat,D5: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C4 @ D5 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D5 ) ) ) ).

% insert_mono
thf(fact_4999_insert__mono,axiom,
    ! [C4: set_VEBT_VEBT,D5: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ C4 @ D5 )
     => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ A @ C4 ) @ ( insert_VEBT_VEBT @ A @ D5 ) ) ) ).

% insert_mono
thf(fact_5000_insert__mono,axiom,
    ! [C4: set_real,D5: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C4 @ D5 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C4 ) @ ( insert_real @ A @ D5 ) ) ) ).

% insert_mono
thf(fact_5001_insert__mono,axiom,
    ! [C4: set_int,D5: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C4 @ D5 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D5 ) ) ) ).

% insert_mono
thf(fact_5002_subset__insert,axiom,
    ! [X2: nat,A2: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B5 ) )
        = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_5003_subset__insert,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ B5 ) )
        = ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_5004_subset__insert,axiom,
    ! [X2: real,A2: set_real,B5: set_real] :
      ( ~ ( member_real @ X2 @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ B5 ) )
        = ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_5005_subset__insert,axiom,
    ! [X2: int,A2: set_int,B5: set_int] :
      ( ~ ( member_int @ X2 @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X2 @ B5 ) )
        = ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_5006_subset__insertI,axiom,
    ! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).

% subset_insertI
thf(fact_5007_subset__insertI,axiom,
    ! [B5: set_VEBT_VEBT,A: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ B5 @ ( insert_VEBT_VEBT @ A @ B5 ) ) ).

% subset_insertI
thf(fact_5008_subset__insertI,axiom,
    ! [B5: set_real,A: real] : ( ord_less_eq_set_real @ B5 @ ( insert_real @ A @ B5 ) ) ).

% subset_insertI
thf(fact_5009_subset__insertI,axiom,
    ! [B5: set_int,A: int] : ( ord_less_eq_set_int @ B5 @ ( insert_int @ A @ B5 ) ) ).

% subset_insertI
thf(fact_5010_subset__insertI2,axiom,
    ! [A2: set_nat,B5: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_5011_subset__insertI2,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
     => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_5012_subset__insertI2,axiom,
    ! [A2: set_real,B5: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_5013_subset__insertI2,axiom,
    ! [A2: set_int,B5: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_5014_insert__Diff__if,axiom,
    ! [X2: vEBT_VEBT,B5: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X2 @ B5 )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X2 @ A2 ) @ B5 )
          = ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) )
      & ( ~ ( member_VEBT_VEBT @ X2 @ B5 )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X2 @ A2 ) @ B5 )
          = ( insert_VEBT_VEBT @ X2 @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5015_insert__Diff__if,axiom,
    ! [X2: real,B5: set_real,A2: set_real] :
      ( ( ( member_real @ X2 @ B5 )
       => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ B5 )
          = ( minus_minus_set_real @ A2 @ B5 ) ) )
      & ( ~ ( member_real @ X2 @ B5 )
       => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ B5 )
          = ( insert_real @ X2 @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5016_insert__Diff__if,axiom,
    ! [X2: int,B5: set_int,A2: set_int] :
      ( ( ( member_int @ X2 @ B5 )
       => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A2 ) @ B5 )
          = ( minus_minus_set_int @ A2 @ B5 ) ) )
      & ( ~ ( member_int @ X2 @ B5 )
       => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A2 ) @ B5 )
          = ( insert_int @ X2 @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5017_insert__Diff__if,axiom,
    ! [X2: nat,B5: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X2 @ B5 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B5 )
          = ( minus_minus_set_nat @ A2 @ B5 ) ) )
      & ( ~ ( member_nat @ X2 @ B5 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B5 )
          = ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5018_foldr__cong,axiom,
    ! [A: assn,B: assn,L: list_assn,K: list_assn,F: assn > assn > assn,G: assn > assn > assn] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: assn,X3: assn] :
              ( ( member_assn @ X3 @ ( set_assn2 @ L ) )
             => ( ( F @ X3 @ A3 )
                = ( G @ X3 @ A3 ) ) )
         => ( ( foldr_assn_assn @ F @ L @ A )
            = ( foldr_assn_assn @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_5019_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_o,K: list_o,F: $o > nat > nat,G: $o > nat > nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: nat,X3: $o] :
              ( ( member_o @ X3 @ ( set_o2 @ L ) )
             => ( ( F @ X3 @ A3 )
                = ( G @ X3 @ A3 ) ) )
         => ( ( foldr_o_nat @ F @ L @ A )
            = ( foldr_o_nat @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_5020_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_nat,K: list_nat,F: nat > nat > nat,G: nat > nat > nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: nat,X3: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ L ) )
             => ( ( F @ X3 @ A3 )
                = ( G @ X3 @ A3 ) ) )
         => ( ( foldr_nat_nat @ F @ L @ A )
            = ( foldr_nat_nat @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_5021_foldr__cong,axiom,
    ! [A: real,B: real,L: list_real,K: list_real,F: real > real > real,G: real > real > real] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A3: real,X3: real] :
              ( ( member_real @ X3 @ ( set_real2 @ L ) )
             => ( ( F @ X3 @ A3 )
                = ( G @ X3 @ A3 ) ) )
         => ( ( foldr_real_real @ F @ L @ A )
            = ( foldr_real_real @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_5022_Collect__conv__if,axiom,
    ! [P: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
thf(fact_5023_Collect__conv__if,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_5024_Collect__conv__if,axiom,
    ! [P: product_prod_int_int > $o,A: product_prod_int_int] :
      ( ( ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if
thf(fact_5025_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_5026_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_5027_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_5028_Collect__conv__if2,axiom,
    ! [P: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
thf(fact_5029_Collect__conv__if2,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_5030_Collect__conv__if2,axiom,
    ! [P: product_prod_int_int > $o,A: product_prod_int_int] :
      ( ( ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if2
thf(fact_5031_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_5032_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_5033_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_5034_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5035_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5036_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5037_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5038_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5039_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5040_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_5041_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_5042_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_5043_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_5044_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_5045_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_5046_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = A )
      = ( ( divide_divide_nat @ A @ B )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_5047_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = A )
      = ( ( divide_divide_int @ A @ B )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_5048_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_5049_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D3: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_5050_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D3: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_5051_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_5052_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_5053_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_5054_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_5055_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P4: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P4 )
       => ( ( ord_less_nat @ M @ P4 )
         => ( ! [N4: nat] :
                ( ( ord_less_nat @ N4 @ P4 )
               => ( ( P @ N4 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P4 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_5056_nat__mod__lem,axiom,
    ! [N2: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ B @ N2 )
        = ( ( modulo_modulo_nat @ B @ N2 )
          = B ) ) ) ).

% nat_mod_lem
thf(fact_5057_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_5058_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_5059_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N2: nat] :
      ( ( ( plus_plus_nat @ L @ K )
        = ( plus_plus_nat @ D @ ( modulo_modulo_nat @ K @ L ) ) )
     => ( ( ord_less_nat @ N2 @ L )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ D @ N2 ) @ L )
          = N2 ) ) ) ).

% word_rot_lem
thf(fact_5060_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_5061_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M7: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M7 @ N ) @ M7 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M7 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_5062_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_5063_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_5064_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_5065_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_5066_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_5067_msrevs_I2_J,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) @ N2 )
      = ( modulo_modulo_nat @ M @ N2 ) ) ).

% msrevs(2)
thf(fact_5068_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_5069_subset__singletonD,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) )
     => ( ( A2 = bot_bo8194388402131092736T_VEBT )
        | ( A2
          = ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singletonD
thf(fact_5070_subset__singletonD,axiom,
    ! [A2: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
     => ( ( A2 = bot_bot_set_nat )
        | ( A2
          = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_5071_subset__singletonD,axiom,
    ! [A2: set_real,X2: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) )
     => ( ( A2 = bot_bot_set_real )
        | ( A2
          = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_5072_subset__singletonD,axiom,
    ! [A2: set_int,X2: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_5073_subset__singleton__iff,axiom,
    ! [X8: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ X8 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( X8 = bot_bo8194388402131092736T_VEBT )
        | ( X8
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singleton_iff
thf(fact_5074_subset__singleton__iff,axiom,
    ! [X8: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X8 = bot_bot_set_nat )
        | ( X8
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_5075_subset__singleton__iff,axiom,
    ! [X8: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X8 = bot_bot_set_real )
        | ( X8
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_5076_subset__singleton__iff,axiom,
    ! [X8: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X8 = bot_bot_set_int )
        | ( X8
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_5077_Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B5 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Diff_insert
thf(fact_5078_Diff__insert,axiom,
    ! [A2: set_int,A: int,B5: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B5 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_5079_Diff__insert,axiom,
    ! [A2: set_real,A: real,B5: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B5 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B5 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_5080_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_5081_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A2 )
     => ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5082_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5083_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5084_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5085_Diff__insert2,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B5 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_5086_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B5: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B5 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_5087_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B5: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B5 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_5088_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_5089_insert__minus__eq,axiom,
    ! [X2: vEBT_VEBT,Y2: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( X2 != Y2 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X2 @ A2 ) @ ( insert_VEBT_VEBT @ Y2 @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ X2 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ Y2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5090_insert__minus__eq,axiom,
    ! [X2: int,Y2: int,A2: set_int] :
      ( ( X2 != Y2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A2 ) @ ( insert_int @ Y2 @ bot_bot_set_int ) )
        = ( insert_int @ X2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5091_insert__minus__eq,axiom,
    ! [X2: real,Y2: real,A2: set_real] :
      ( ( X2 != Y2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ ( insert_real @ Y2 @ bot_bot_set_real ) )
        = ( insert_real @ X2 @ ( minus_minus_set_real @ A2 @ ( insert_real @ Y2 @ bot_bot_set_real ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5092_insert__minus__eq,axiom,
    ! [X2: nat,Y2: nat,A2: set_nat] :
      ( ( X2 != Y2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) )
        = ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5093_Diff__insert__absorb,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X2 @ A2 ) @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5094_Diff__insert__absorb,axiom,
    ! [X2: int,A2: set_int] :
      ( ~ ( member_int @ X2 @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A2 ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5095_Diff__insert__absorb,axiom,
    ! [X2: real,A2: set_real] :
      ( ~ ( member_real @ X2 @ A2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5096_Diff__insert__absorb,axiom,
    ! [X2: nat,A2: set_nat] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5097_set__minus__singleton__eq,axiom,
    ! [X2: vEBT_VEBT,X8: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X2 @ X8 )
     => ( ( minus_5127226145743854075T_VEBT @ X8 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_5098_set__minus__singleton__eq,axiom,
    ! [X2: int,X8: set_int] :
      ( ~ ( member_int @ X2 @ X8 )
     => ( ( minus_minus_set_int @ X8 @ ( insert_int @ X2 @ bot_bot_set_int ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_5099_set__minus__singleton__eq,axiom,
    ! [X2: real,X8: set_real] :
      ( ~ ( member_real @ X2 @ X8 )
     => ( ( minus_minus_set_real @ X8 @ ( insert_real @ X2 @ bot_bot_set_real ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_5100_set__minus__singleton__eq,axiom,
    ! [X2: nat,X8: set_nat] :
      ( ~ ( member_nat @ X2 @ X8 )
     => ( ( minus_minus_set_nat @ X8 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_5101_subset__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,X2: vEBT_VEBT,C4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B5 @ ( insert_VEBT_VEBT @ X2 @ C4 ) ) )
      = ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B5 @ C4 ) )
        & ~ ( member_VEBT_VEBT @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5102_subset__Diff__insert,axiom,
    ! [A2: set_real,B5: set_real,X2: real,C4: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B5 @ ( insert_real @ X2 @ C4 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B5 @ C4 ) )
        & ~ ( member_real @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5103_subset__Diff__insert,axiom,
    ! [A2: set_nat,B5: set_nat,X2: nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X2 @ C4 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B5 @ C4 ) )
        & ~ ( member_nat @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5104_subset__Diff__insert,axiom,
    ! [A2: set_int,B5: set_int,X2: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B5 @ ( insert_int @ X2 @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B5 @ C4 ) )
        & ~ ( member_int @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5105_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5106_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5107_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5108_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5109_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5110_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5111_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_5112_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_5113_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_5114_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_5115_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_5116_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_5117_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_5118_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_5119_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_5120_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_5121_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_5122_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_5123_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_5124_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_5125_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_5126_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_5127_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_5128_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_5129_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_5130_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_5131_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_5132_mult__div__mod__eq,axiom,
    ! [B: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ B @ ( divide1791077408188789448l_num1 @ A @ B ) ) @ ( modulo1504961113040953224l_num1 @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5133_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5134_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5135_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5136_mult__div__mod__eq,axiom,
    ! [B: uint32,A: uint32] :
      ( ( plus_plus_uint32 @ ( times_times_uint32 @ B @ ( divide_divide_uint32 @ A @ B ) ) @ ( modulo_modulo_uint32 @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5137_mod__mult__div__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A @ B ) @ ( times_7065122842183080059l_num1 @ B @ ( divide1791077408188789448l_num1 @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5138_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5139_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5140_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5141_mod__mult__div__eq,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ B ) @ ( times_times_uint32 @ B @ ( divide_divide_uint32 @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5142_mod__div__mult__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A @ B ) @ ( times_7065122842183080059l_num1 @ ( divide1791077408188789448l_num1 @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5143_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5144_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5145_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5146_mod__div__mult__eq,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ B ) @ ( times_times_uint32 @ ( divide_divide_uint32 @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5147_div__mult__mod__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( divide1791077408188789448l_num1 @ A @ B ) @ B ) @ ( modulo1504961113040953224l_num1 @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5148_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5149_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5150_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5151_div__mult__mod__eq,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A @ B ) @ B ) @ ( modulo_modulo_uint32 @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5152_mod__div__decomp,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( A
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( divide1791077408188789448l_num1 @ A @ B ) @ B ) @ ( modulo1504961113040953224l_num1 @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5153_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5154_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5155_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5156_mod__div__decomp,axiom,
    ! [A: uint32,B: uint32] :
      ( A
      = ( plus_plus_uint32 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A @ B ) @ B ) @ ( modulo_modulo_uint32 @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5157_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5158_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5159_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5160_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5161_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5162_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5163_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5164_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5165_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5166_minus__mult__div__eq__mod,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A @ ( times_7065122842183080059l_num1 @ B @ ( divide1791077408188789448l_num1 @ A @ B ) ) )
      = ( modulo1504961113040953224l_num1 @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5167_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5168_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5169_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5170_minus__mult__div__eq__mod,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ A @ ( times_times_uint32 @ B @ ( divide_divide_uint32 @ A @ B ) ) )
      = ( modulo_modulo_uint32 @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5171_minus__mod__eq__mult__div,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A @ ( modulo1504961113040953224l_num1 @ A @ B ) )
      = ( times_7065122842183080059l_num1 @ B @ ( divide1791077408188789448l_num1 @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5172_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5173_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5174_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5175_minus__mod__eq__mult__div,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ A @ ( modulo_modulo_uint32 @ A @ B ) )
      = ( times_times_uint32 @ B @ ( divide_divide_uint32 @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5176_minus__mod__eq__div__mult,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A @ ( modulo1504961113040953224l_num1 @ A @ B ) )
      = ( times_7065122842183080059l_num1 @ ( divide1791077408188789448l_num1 @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5177_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5178_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5179_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5180_minus__mod__eq__div__mult,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ A @ ( modulo_modulo_uint32 @ A @ B ) )
      = ( times_times_uint32 @ ( divide_divide_uint32 @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5181_minus__div__mult__eq__mod,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A @ ( times_7065122842183080059l_num1 @ ( divide1791077408188789448l_num1 @ A @ B ) @ B ) )
      = ( modulo1504961113040953224l_num1 @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5182_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5183_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5184_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5185_minus__div__mult__eq__mod,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ A @ ( times_times_uint32 @ ( divide_divide_uint32 @ A @ B ) @ B ) )
      = ( modulo_modulo_uint32 @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5186_zmde,axiom,
    ! [B: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ B @ ( divide1791077408188789448l_num1 @ A @ B ) )
      = ( minus_4019991460397169231l_num1 @ A @ ( modulo1504961113040953224l_num1 @ A @ B ) ) ) ).

% zmde
thf(fact_5187_zmde,axiom,
    ! [B: int,A: int] :
      ( ( times_times_int @ B @ ( divide_divide_int @ A @ B ) )
      = ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% zmde
thf(fact_5188_zmde,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% zmde
thf(fact_5189_zmde,axiom,
    ! [B: uint32,A: uint32] :
      ( ( times_times_uint32 @ B @ ( divide_divide_uint32 @ A @ B ) )
      = ( minus_minus_uint32 @ A @ ( modulo_modulo_uint32 @ A @ B ) ) ) ).

% zmde
thf(fact_5190_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_5191_div__less__mono,axiom,
    ! [A2: nat,B5: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A2 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B5 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_5192_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_5193_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_5194_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 )
       => ~ ! [S: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_5195_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 )
       => ~ ! [S: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_5196_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_5197_div__mod__decomp,axiom,
    ! [A2: nat,N2: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_5198_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M7: nat,N: nat] : ( minus_minus_nat @ M7 @ ( times_times_nat @ ( divide_divide_nat @ M7 @ N ) @ N ) ) ) ) ).

% modulo_nat_def
thf(fact_5199_subset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ B5 ) )
      = ( ( ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) @ B5 ) )
        & ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_5200_subset__insert__iff,axiom,
    ! [A2: set_real,X2: real,B5: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ B5 ) )
      = ( ( ( member_real @ X2 @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B5 ) )
        & ( ~ ( member_real @ X2 @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_5201_subset__insert__iff,axiom,
    ! [A2: set_nat,X2: nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B5 ) )
      = ( ( ( member_nat @ X2 @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 ) )
        & ( ~ ( member_nat @ X2 @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_5202_subset__insert__iff,axiom,
    ! [A2: set_int,X2: int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X2 @ B5 ) )
      = ( ( ( member_int @ X2 @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B5 ) )
        & ( ~ ( member_int @ X2 @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_5203_Diff__single__insert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) @ B5 )
     => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_5204_Diff__single__insert,axiom,
    ! [A2: set_real,X2: real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B5 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X2 @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_5205_Diff__single__insert,axiom,
    ! [A2: set_nat,X2: nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_5206_Diff__single__insert,axiom,
    ! [A2: set_int,X2: int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B5 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X2 @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_5207_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_5208_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_5209_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_5210_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_5211_remove__subset,axiom,
    ! [X2: vEBT_VEBT,S2: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ S2 )
     => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5212_remove__subset,axiom,
    ! [X2: int,S2: set_int] :
      ( ( member_int @ X2 @ S2 )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5213_remove__subset,axiom,
    ! [X2: real,S2: set_real] :
      ( ( member_real @ X2 @ S2 )
     => ( ord_less_set_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5214_remove__subset,axiom,
    ! [X2: nat,S2: set_nat] :
      ( ( member_nat @ X2 @ S2 )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5215_set__update__subset__insert,axiom,
    ! [Xs2: list_nat,I: nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X2 ) ) @ ( insert_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_5216_set__update__subset__insert,axiom,
    ! [Xs2: list_real,I: nat,X2: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X2 ) ) @ ( insert_real @ X2 @ ( set_real2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_5217_set__update__subset__insert,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,I: nat,X2: produc6575502325842934193n_assn] : ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ Xs2 @ I @ X2 ) ) @ ( insert5290817439147925377n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_5218_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) ) @ ( insert_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_5219_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X2: vEBT_VEBTi] : ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) ) @ ( insert_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_5220_set__update__subset__insert,axiom,
    ! [Xs2: list_int,I: nat,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X2 ) ) @ ( insert_int @ X2 @ ( set_int2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_5221_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_5222_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_5223_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_5224_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5225_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5226_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5227_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_5228_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_5229_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 )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_5230_mod__lemma,axiom,
    ! [C: nat,R3: nat,B: nat,Q2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ R3 @ B )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ Q2 @ C ) ) @ R3 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mod_lemma
thf(fact_5231_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] :
      ( ( foldr_real_nat
        @ ^ [X: real] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_real @ L ) ) ) ).

% foldr_length_aux
thf(fact_5232_foldr__length__aux,axiom,
    ! [L: list_o,A: nat] :
      ( ( foldr_o_nat
        @ ^ [X: $o] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_o @ L ) ) ) ).

% foldr_length_aux
thf(fact_5233_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] :
      ( ( foldr_nat_nat
        @ ^ [X: nat] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_nat @ L ) ) ) ).

% foldr_length_aux
thf(fact_5234_foldr__length__aux,axiom,
    ! [L: list_int,A: nat] :
      ( ( foldr_int_nat
        @ ^ [X: int] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_int @ L ) ) ) ).

% foldr_length_aux
thf(fact_5235_psubset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ B5 ) )
      = ( ( ( member_VEBT_VEBT @ X2 @ B5 )
         => ( ord_le3480810397992357184T_VEBT @ A2 @ B5 ) )
        & ( ~ ( member_VEBT_VEBT @ X2 @ B5 )
         => ( ( ( member_VEBT_VEBT @ X2 @ A2 )
             => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) @ B5 ) )
            & ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
             => ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5236_psubset__insert__iff,axiom,
    ! [A2: set_real,X2: real,B5: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real @ X2 @ B5 ) )
      = ( ( ( member_real @ X2 @ B5 )
         => ( ord_less_set_real @ A2 @ B5 ) )
        & ( ~ ( member_real @ X2 @ B5 )
         => ( ( ( member_real @ X2 @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B5 ) )
            & ( ~ ( member_real @ X2 @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5237_psubset__insert__iff,axiom,
    ! [A2: set_nat,X2: nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X2 @ B5 ) )
      = ( ( ( member_nat @ X2 @ B5 )
         => ( ord_less_set_nat @ A2 @ B5 ) )
        & ( ~ ( member_nat @ X2 @ B5 )
         => ( ( ( member_nat @ X2 @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 ) )
            & ( ~ ( member_nat @ X2 @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5238_psubset__insert__iff,axiom,
    ! [A2: set_int,X2: int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X2 @ B5 ) )
      = ( ( ( member_int @ X2 @ B5 )
         => ( ord_less_set_int @ A2 @ B5 ) )
        & ( ~ ( member_int @ X2 @ B5 )
         => ( ( ( member_int @ X2 @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B5 ) )
            & ( ~ ( member_int @ X2 @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5239_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_5240_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5241_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5242_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5243_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_5244_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_5245_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_5246_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_5247_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_5248_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_5249_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_5250_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_5251_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
     => ( ( insert5290817439147925377n_assn @ ( nth_Pr1769885009046257848n_assn @ L @ I ) @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ X2 ) ) )
        = ( insert5290817439147925377n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5252_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ L @ I ) @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ X2 ) ) )
        = ( insert_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5253_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ L @ I ) @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ X2 ) ) )
        = ( insert_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5254_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,X2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( insert_real @ ( nth_real @ L @ I ) @ ( set_real2 @ ( list_update_real @ L @ I @ X2 ) ) )
        = ( insert_real @ X2 @ ( set_real2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5255_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( insert_o @ ( nth_o @ L @ I ) @ ( set_o2 @ ( list_update_o @ L @ I @ X2 ) ) )
        = ( insert_o @ X2 @ ( set_o2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5256_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( insert_nat @ ( nth_nat @ L @ I ) @ ( set_nat2 @ ( list_update_nat @ L @ I @ X2 ) ) )
        = ( insert_nat @ X2 @ ( set_nat2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5257_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( insert_int @ ( nth_int @ L @ I ) @ ( set_int2 @ ( list_update_int @ L @ I @ X2 ) ) )
        = ( insert_int @ X2 @ ( set_int2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5258_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5259_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5260_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5261_bits__stable__imp__add__self,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( ( divide1791077408188789448l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p361126936061061375l_num1 @ A @ ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% bits_stable_imp_add_self
thf(fact_5262_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_5263_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_5264_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_5265_bits__stable__imp__add__self,axiom,
    ! [A: uint32] :
      ( ( ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_uint32 @ A @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
        = zero_zero_uint32 ) ) ).

% bits_stable_imp_add_self
thf(fact_5266_div__exp__mod__exp__eq,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat,M: nat] :
      ( ( modulo1504961113040953224l_num1 @ ( divide1791077408188789448l_num1 @ A @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) )
      = ( divide1791077408188789448l_num1 @ ( modulo1504961113040953224l_num1 @ A @ ( 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_5267_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5268_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5269_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5270_div__exp__mod__exp__eq,axiom,
    ! [A: uint32,N2: nat,M: nat] :
      ( ( modulo_modulo_uint32 @ ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A @ ( 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_5271_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_5272_verit__le__mono__div,axiom,
    ! [A2: nat,B5: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B5 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_5273_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5274_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5275_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5276_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo1504961113040953224l_num1 @ ( times_7065122842183080059l_num1 @ A @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_7065122842183080059l_num1 @ ( modulo1504961113040953224l_num1 @ A @ ( 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_5277_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5278_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5279_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5280_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: uint32] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_uint32 @ ( times_times_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_uint32 @ ( modulo_modulo_uint32 @ A @ ( 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_5281_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_5282_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_5283_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_5284_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5285_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5286_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5287_set__bit__Suc,axiom,
    ! [N2: nat,A: word_N3645301735248828278l_num1] :
      ( ( bit_se4894374433684937756l_num1 @ ( suc @ N2 ) @ A )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4894374433684937756l_num1 @ N2 @ ( divide1791077408188789448l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5288_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5289_set__bit__Suc,axiom,
    ! [N2: nat,A: uint32] :
      ( ( bit_se6647067497041451410uint32 @ ( suc @ N2 ) @ A )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ N2 @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5290_set__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5291_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5292_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5293_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5294_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5295_norm__divide__numeral,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_5296_norm__divide__numeral,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_5297_norm__mult__numeral2,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_5298_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_5299_norm__mult__numeral1,axiom,
    ! [W: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_5300_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_5301_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_5302_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_5303_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_5304_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5305_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5306_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_5307_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_5308_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_5309_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_5310_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_5311_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_5312_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_5313_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_5314_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_5315_mod__plus__cong,axiom,
    ! [B: int,B3: int,X2: int,X7: int,Y2: int,Y6: int,Z6: int] :
      ( ( B = B3 )
     => ( ( ( modulo_modulo_int @ X2 @ B3 )
          = ( modulo_modulo_int @ X7 @ B3 ) )
       => ( ( ( modulo_modulo_int @ Y2 @ B3 )
            = ( modulo_modulo_int @ Y6 @ B3 ) )
         => ( ( ( plus_plus_int @ X7 @ Y6 )
              = Z6 )
           => ( ( modulo_modulo_int @ ( plus_plus_int @ X2 @ Y2 ) @ B )
              = ( modulo_modulo_int @ Z6 @ B3 ) ) ) ) ) ) ).

% mod_plus_cong
thf(fact_5316_zmod__helper,axiom,
    ! [N2: int,M: int,K: int,A: int] :
      ( ( ( modulo_modulo_int @ N2 @ M )
        = K )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ N2 @ A ) @ M )
        = ( modulo_modulo_int @ ( plus_plus_int @ K @ A ) @ M ) ) ) ).

% zmod_helper
thf(fact_5317_Word_Omod__minus__cong,axiom,
    ! [B: int,B3: int,X2: int,X7: int,Y2: int,Y6: int,Z6: int] :
      ( ( B = B3 )
     => ( ( ( modulo_modulo_int @ X2 @ B3 )
          = ( modulo_modulo_int @ X7 @ B3 ) )
       => ( ( ( modulo_modulo_int @ Y2 @ B3 )
            = ( modulo_modulo_int @ Y6 @ B3 ) )
         => ( ( ( minus_minus_int @ X7 @ Y6 )
              = Z6 )
           => ( ( modulo_modulo_int @ ( minus_minus_int @ X2 @ Y2 ) @ B )
              = ( modulo_modulo_int @ Z6 @ B3 ) ) ) ) ) ) ).

% Word.mod_minus_cong
thf(fact_5318_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_5319_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_5320_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_5321_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_5322_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q5: int] :
            ( M
            = ( times_times_int @ D @ Q5 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_5323_int__mod__ge,axiom,
    ! [A: int,N2: int] :
      ( ( ord_less_int @ A @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ A @ ( modulo_modulo_int @ A @ N2 ) ) ) ) ).

% int_mod_ge
thf(fact_5324_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_5325_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_5326_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_5327_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_5328_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_5329_int__mod__eq,axiom,
    ! [B: int,N2: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ N2 )
       => ( ( ( modulo_modulo_int @ A @ N2 )
            = ( modulo_modulo_int @ B @ N2 ) )
         => ( ( modulo_modulo_int @ A @ N2 )
            = B ) ) ) ) ).

% int_mod_eq
thf(fact_5330_int__mod__lem,axiom,
    ! [N2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ N2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ B )
          & ( ord_less_int @ B @ N2 ) )
        = ( ( modulo_modulo_int @ B @ N2 )
          = B ) ) ) ).

% int_mod_lem
thf(fact_5331_int__mod__le_H,axiom,
    ! [B: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ B @ N2 ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ B @ N2 ) @ ( minus_minus_int @ B @ N2 ) ) ) ).

% int_mod_le'
thf(fact_5332_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
          & ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% nonneg_mod_div
thf(fact_5333_zdiv__mono__strict,axiom,
    ! [A2: int,B5: int,N2: int] :
      ( ( ord_less_int @ A2 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A2 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B5 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B5 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_5334_div__mod__decomp__int,axiom,
    ! [A2: int,N2: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_5335_mod__div__equality__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ B )
      = ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_equality_div_eq
thf(fact_5336_word__mod__div__equality,axiom,
    ! [N2: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( divide1791077408188789448l_num1 @ N2 @ B ) @ B ) @ ( modulo1504961113040953224l_num1 @ N2 @ B ) )
      = N2 ) ).

% word_mod_div_equality
thf(fact_5337_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_5338_pos__mod__bound2,axiom,
    ! [A: int] : ( ord_less_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% pos_mod_bound2
thf(fact_5339_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_5340_int__mod__ge_H,axiom,
    ! [B: int,N2: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ B @ N2 ) @ ( modulo_modulo_int @ B @ N2 ) ) ) ) ).

% int_mod_ge'
thf(fact_5341_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_5342_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_5343_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_5344_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_5345_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_5346_pos__mod__sign2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% pos_mod_sign2
thf(fact_5347_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_5348_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_5349_mod__exp__less__eq__exp,axiom,
    ! [A: int,N2: nat] : ( ord_less_int @ ( modulo_modulo_int @ A @ ( 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_5350_mod__power__lem,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( ord_less_eq_nat @ M @ N2 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ M ) )
            = zero_zero_int ) )
        & ( ~ ( ord_less_eq_nat @ M @ N2 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ M ) )
            = ( power_power_int @ A @ N2 ) ) ) ) ) ).

% mod_power_lem
thf(fact_5351_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 )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_5352_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_5353_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_5354_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_5355_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_5356_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_5357_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_5358_z1pmod2,axiom,
    ! [B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = one_one_int ) ).

% z1pmod2
thf(fact_5359_verit__le__mono__div__int,axiom,
    ! [A2: int,B5: int,N2: int] :
      ( ( ord_less_int @ A2 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B5 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B5 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_5360_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 ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_5361_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 ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_5362_norm__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_5363_norm__minus__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_5364_p1mod22k_H,axiom,
    ! [B: int,N2: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( 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 @ B @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% p1mod22k'
thf(fact_5365_p1mod22k,axiom,
    ! [B: int,N2: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ 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 @ B @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ one_one_int ) ) ).

% p1mod22k
thf(fact_5366_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_5367_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A @ ( 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 @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem
thf(fact_5368_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_5369_norm__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y2 ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).

% norm_mult
thf(fact_5370_norm__mult,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y2 ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).

% norm_mult
thf(fact_5371_norm__ge__zero,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% norm_ge_zero
thf(fact_5372_norm__divide,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_divide
thf(fact_5373_norm__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_divide
thf(fact_5374_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_5375_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_5376_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_5377_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_5378_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_5379_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_5380_norm__mult__less,axiom,
    ! [X2: real,R3: real,Y2: real,S3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R3 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y2 ) ) @ ( times_times_real @ R3 @ S3 ) ) ) ) ).

% norm_mult_less
thf(fact_5381_norm__mult__less,axiom,
    ! [X2: complex,R3: real,Y2: complex,S3: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R3 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y2 ) ) @ ( times_times_real @ R3 @ S3 ) ) ) ) ).

% norm_mult_less
thf(fact_5382_norm__mult__ineq,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y2 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).

% norm_mult_ineq
thf(fact_5383_norm__mult__ineq,axiom,
    ! [X2: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y2 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).

% norm_mult_ineq
thf(fact_5384_norm__add__less,axiom,
    ! [X2: real,R3: real,Y2: real,S3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R3 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_real @ R3 @ S3 ) ) ) ) ).

% norm_add_less
thf(fact_5385_norm__add__less,axiom,
    ! [X2: complex,R3: real,Y2: complex,S3: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R3 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ ( plus_plus_real @ R3 @ S3 ) ) ) ) ).

% norm_add_less
thf(fact_5386_norm__triangle__lt,axiom,
    ! [X2: real,Y2: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_5387_norm__triangle__lt,axiom,
    ! [X2: complex,Y2: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_5388_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_5389_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_5390_norm__triangle__le,axiom,
    ! [X2: real,Y2: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_5391_norm__triangle__le,axiom,
    ! [X2: complex,Y2: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_5392_norm__triangle__ineq,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).

% norm_triangle_ineq
thf(fact_5393_norm__triangle__ineq,axiom,
    ! [X2: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).

% norm_triangle_ineq
thf(fact_5394_norm__triangle__mono,axiom,
    ! [A: real,R3: real,B: real,S3: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R3 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S3 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R3 @ S3 ) ) ) ) ).

% norm_triangle_mono
thf(fact_5395_norm__triangle__mono,axiom,
    ! [A: complex,R3: real,B: complex,S3: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R3 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S3 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R3 @ S3 ) ) ) ) ).

% norm_triangle_mono
thf(fact_5396_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_5397_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_5398_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_5399_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_5400_norm__triangle__le__diff,axiom,
    ! [X2: real,Y2: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_5401_norm__triangle__le__diff,axiom,
    ! [X2: complex,Y2: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_5402_norm__diff__triangle__le,axiom,
    ! [X2: real,Y2: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_5403_norm__diff__triangle__le,axiom,
    ! [X2: complex,Y2: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_5404_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_5405_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_5406_norm__triangle__sub,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) ) ) ).

% norm_triangle_sub
thf(fact_5407_norm__triangle__sub,axiom,
    ! [X2: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) ) ) ).

% norm_triangle_sub
thf(fact_5408_norm__diff__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_5409_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_5410_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_5411_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_5412_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_5413_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_5414_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5415_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5416_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_5417_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_5418_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_5419_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_5420_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_5421_vebt__inserti_Osimps,axiom,
    ( vEBT_vebt_inserti
    = ( ^ [T2: vEBT_VEBTi,X: nat] :
          ( vEBT_c6028912655521741485_VEBTi
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
              @ ^ [Minma: product_prod_nat_nat] :
                  ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                  @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [L2: nat] :
                                        ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [H: nat] :
                                            ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                            @ ^ [Len: nat] :
                                                ( if_Hea8453224502484754311_VEBTi
                                                @ ( ( ord_less_nat @ H @ Len )
                                                  & ~ ( ( X = Mi3 )
                                                      | ( X = Ma3 ) ) )
                                                @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                  @ ^ [Node: vEBT_VEBTi] :
                                                      ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                      @ ^ [Empt: $o] :
                                                          ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_inserti @ Node @ L2 )
                                                          @ ^ [Newnode2: vEBT_VEBTi] :
                                                              ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                              @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                  ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( vEBT_vebt_inserti @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                  @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A4: $o,B4: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B4 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
          @ T2 ) ) ) ).

% vebt_inserti.simps
thf(fact_5422_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ 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_5423_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_5424_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_5425_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_5426_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_5427_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_5428_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_5429_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs: list_nat] : Xs ) ) ).

% map_ident
thf(fact_5430_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_5431_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_5432_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_5433_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_5434_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_5435_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_5436_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_5437_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_5438_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_5439_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_5440_map__eq__conv,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Xs2 ) )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_5441_map__eq__conv,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Xs2 ) )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_5442_map__eq__conv,axiom,
    ! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F @ Xs2 )
        = ( map_nat_nat @ G @ Xs2 ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_5443_map__eq__conv,axiom,
    ! [F: nat > $o,Xs2: list_nat,G: nat > $o] :
      ( ( ( map_nat_o @ F @ Xs2 )
        = ( map_nat_o @ G @ Xs2 ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_5444_map__eq__conv,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs2: list_P8527749157015355191n_assn,G: produc6575502325842934193n_assn > assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Xs2 )
        = ( map_Pr8991440229025900053n_assn @ G @ Xs2 ) )
      = ( ! [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_5445_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_5446_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_5447_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_5448_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_5449_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_5450_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_5451_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_5452_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_5453_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_5454_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_5455_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_5456_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_5457_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_5458_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_5459_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_5460_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_5461_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_5462_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_5463_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_5464_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_5465_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_5466_list_Omap__cong,axiom,
    ! [X2: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( X2 = Ya )
     => ( ! [Z3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z3 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F @ X2 )
          = ( map_VEBT_VEBT_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5467_list_Omap__cong,axiom,
    ! [X2: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( X2 = Ya )
     => ( ! [Z3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z3 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_VEBT_VEBT_real @ F @ X2 )
          = ( map_VEBT_VEBT_real @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5468_list_Omap__cong,axiom,
    ! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
      ( ( X2 = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_nat_nat @ F @ X2 )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5469_list_Omap__cong,axiom,
    ! [X2: list_nat,Ya: list_nat,F: nat > $o,G: nat > $o] :
      ( ( X2 = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_nat_o @ F @ X2 )
          = ( map_nat_o @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5470_list_Omap__cong,axiom,
    ! [X2: list_P8527749157015355191n_assn,Ya: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn,G: produc6575502325842934193n_assn > assn] :
      ( ( X2 = Ya )
     => ( ! [Z3: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ Z3 @ ( set_Pr1139785259514867910n_assn @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_Pr8991440229025900053n_assn @ F @ X2 )
          = ( map_Pr8991440229025900053n_assn @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5471_list_Omap__cong0,axiom,
    ! [X2: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [Z3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z3 @ ( set_VEBT_VEBT2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ X2 )
        = ( map_VEBT_VEBT_nat @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_5472_list_Omap__cong0,axiom,
    ! [X2: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [Z3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z3 @ ( set_VEBT_VEBT2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ X2 )
        = ( map_VEBT_VEBT_real @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_5473_list_Omap__cong0,axiom,
    ! [X2: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_nat_nat @ F @ X2 )
        = ( map_nat_nat @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_5474_list_Omap__cong0,axiom,
    ! [X2: list_nat,F: nat > $o,G: nat > $o] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_nat_o @ F @ X2 )
        = ( map_nat_o @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_5475_list_Omap__cong0,axiom,
    ! [X2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn,G: produc6575502325842934193n_assn > assn] :
      ( ! [Z3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ Z3 @ ( set_Pr1139785259514867910n_assn @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_Pr8991440229025900053n_assn @ F @ X2 )
        = ( map_Pr8991440229025900053n_assn @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_5476_list_Oinj__map__strong,axiom,
    ! [X2: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F: vEBT_VEBT > nat,Fa: vEBT_VEBT > nat] :
      ( ! [Z3: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z3 @ ( set_VEBT_VEBT2 @ X2 ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_nat @ F @ X2 )
          = ( map_VEBT_VEBT_nat @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_5477_list_Oinj__map__strong,axiom,
    ! [X2: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F: vEBT_VEBT > real,Fa: vEBT_VEBT > real] :
      ( ! [Z3: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z3 @ ( set_VEBT_VEBT2 @ X2 ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_real @ F @ X2 )
          = ( map_VEBT_VEBT_real @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_5478_list_Oinj__map__strong,axiom,
    ! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F @ X2 )
          = ( map_nat_nat @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_5479_list_Oinj__map__strong,axiom,
    ! [X2: list_nat,Xa: list_nat,F: nat > $o,Fa: nat > $o] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_nat_o @ F @ X2 )
          = ( map_nat_o @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_5480_list_Oinj__map__strong,axiom,
    ! [X2: list_P8527749157015355191n_assn,Xa: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn,Fa: produc6575502325842934193n_assn > assn] :
      ( ! [Z3: produc6575502325842934193n_assn,Za: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ Z3 @ ( set_Pr1139785259514867910n_assn @ X2 ) )
         => ( ( member7957490590177025114n_assn @ Za @ ( set_Pr1139785259514867910n_assn @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_Pr8991440229025900053n_assn @ F @ X2 )
          = ( map_Pr8991440229025900053n_assn @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_5481_map__ext,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_5482_map__ext,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_5483_map__ext,axiom,
    ! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_nat @ F @ Xs2 )
        = ( map_nat_nat @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_5484_map__ext,axiom,
    ! [Xs2: list_nat,F: nat > $o,G: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_o @ F @ Xs2 )
        = ( map_nat_o @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_5485_map__ext,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn,G: produc6575502325842934193n_assn > assn] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_Pr8991440229025900053n_assn @ F @ Xs2 )
        = ( map_Pr8991440229025900053n_assn @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_5486_map__idI,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_VE8901447254227204932T_VEBT @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5487_map__idI,axiom,
    ! [Xs2: list_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5488_map__idI,axiom,
    ! [Xs2: list_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_real_real @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5489_map__idI,axiom,
    ! [Xs2: list_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_int_int @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5490_map__idI,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > produc6575502325842934193n_assn] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr7925354932063753860n_assn @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5491_map__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( Xs2 = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
          = ( map_VEBT_VEBT_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5492_map__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( Xs2 = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_VEBT_VEBT_real @ F @ Xs2 )
          = ( map_VEBT_VEBT_real @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5493_map__cong,axiom,
    ! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
      ( ( Xs2 = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_nat @ F @ Xs2 )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5494_map__cong,axiom,
    ! [Xs2: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
      ( ( Xs2 = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_o @ F @ Xs2 )
          = ( map_nat_o @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5495_map__cong,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn,G: produc6575502325842934193n_assn > assn] :
      ( ( Xs2 = Ys )
     => ( ! [X3: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_Pr8991440229025900053n_assn @ F @ Xs2 )
          = ( map_Pr8991440229025900053n_assn @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5496_ex__map__conv,axiom,
    ! [Ys: list_o,F: nat > $o] :
      ( ( ? [Xs: list_nat] :
            ( Ys
            = ( map_nat_o @ F @ Xs ) ) )
      = ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Ys ) )
           => ? [Y: nat] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_5497_ex__map__conv,axiom,
    ! [Ys: list_assn,F: produc6575502325842934193n_assn > assn] :
      ( ( ? [Xs: list_P8527749157015355191n_assn] :
            ( Ys
            = ( map_Pr8991440229025900053n_assn @ F @ Xs ) ) )
      = ( ! [X: assn] :
            ( ( member_assn @ X @ ( set_assn2 @ Ys ) )
           => ? [Y: produc6575502325842934193n_assn] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_5498_ex__map__conv,axiom,
    ! [Ys: list_nat,F: vEBT_VEBT > nat] :
      ( ( ? [Xs: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_nat @ F @ Xs ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ? [Y: vEBT_VEBT] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_5499_ex__map__conv,axiom,
    ! [Ys: list_nat,F: nat > nat] :
      ( ( ? [Xs: list_nat] :
            ( Ys
            = ( map_nat_nat @ F @ Xs ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ? [Y: nat] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_5500_ex__map__conv,axiom,
    ! [Ys: list_real,F: vEBT_VEBT > real] :
      ( ( ? [Xs: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_real @ F @ Xs ) ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Ys ) )
           => ? [Y: vEBT_VEBT] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_5501_map__eq__nth__eq,axiom,
    ! [F: vEBT_VEBT > nat,L: list_VEBT_VEBT,L4: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ L )
        = ( map_VEBT_VEBT_nat @ F @ L4 ) )
     => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
        = ( F @ ( nth_VEBT_VEBT @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_5502_map__eq__nth__eq,axiom,
    ! [F: vEBT_VEBT > real,L: list_VEBT_VEBT,L4: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_real @ F @ L )
        = ( map_VEBT_VEBT_real @ F @ L4 ) )
     => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
        = ( F @ ( nth_VEBT_VEBT @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_5503_map__eq__nth__eq,axiom,
    ! [F: nat > nat,L: list_nat,L4: list_nat,I: nat] :
      ( ( ( map_nat_nat @ F @ L )
        = ( map_nat_nat @ F @ L4 ) )
     => ( ( F @ ( nth_nat @ L @ I ) )
        = ( F @ ( nth_nat @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_5504_map__eq__nth__eq,axiom,
    ! [F: nat > $o,L: list_nat,L4: list_nat,I: nat] :
      ( ( ( map_nat_o @ F @ L )
        = ( map_nat_o @ F @ L4 ) )
     => ( ( F @ ( nth_nat @ L @ I ) )
        = ( F @ ( nth_nat @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_5505_map__eq__nth__eq,axiom,
    ! [F: produc6575502325842934193n_assn > assn,L: list_P8527749157015355191n_assn,L4: list_P8527749157015355191n_assn,I: nat] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ L )
        = ( map_Pr8991440229025900053n_assn @ F @ L4 ) )
     => ( ( F @ ( nth_Pr1769885009046257848n_assn @ L @ I ) )
        = ( F @ ( nth_Pr1769885009046257848n_assn @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_5506_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_5507_time__array__len,axiom,
    ! [P4: array_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_nat @ ( array_len_VEBT_VEBTi @ P4 ) @ H2 )
      = one_one_nat ) ).

% time_array_len
thf(fact_5508_map__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn,X2: produc6575502325842934193n_assn] :
      ( ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
       => ( ( F @ ( nth_Pr1769885009046257848n_assn @ L @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_Pr8991440229025900053n_assn @ F @ ( list_u4534839942911652127n_assn @ L @ I @ X2 ) )
        = ( map_Pr8991440229025900053n_assn @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5509_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > nat,X2: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ ( list_u1324408373059187874T_VEBT @ L @ I @ X2 ) )
        = ( map_VEBT_VEBT_nat @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5510_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > real,X2: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ ( list_u1324408373059187874T_VEBT @ L @ I @ X2 ) )
        = ( map_VEBT_VEBT_real @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5511_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: nat > nat,X2: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( F @ ( nth_nat @ L @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_nat_nat @ F @ ( list_update_nat @ L @ I @ X2 ) )
        = ( map_nat_nat @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5512_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: nat > $o,X2: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( F @ ( nth_nat @ L @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_nat_o @ F @ ( list_update_nat @ L @ I @ X2 ) )
        = ( map_nat_o @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5513_TBOUND__len,axiom,
    ! [Xs2: array_VEBT_VEBTi] : ( time_TBOUND_nat @ ( array_len_VEBT_VEBTi @ Xs2 ) @ one_one_nat ) ).

% TBOUND_len
thf(fact_5514_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( 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 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.cnt'.simps(2)
thf(fact_5515_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X2 )
        = Y2 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y2 != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.cnt'.elims
thf(fact_5516_eq__diff__eq_H,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( X2
        = ( minus_minus_real @ Y2 @ Z ) )
      = ( Y2
        = ( plus_plus_real @ X2 @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_5517_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( 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 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space'.simps(2)
thf(fact_5518_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space2 @ X2 )
        = Y2 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y2
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space'.elims
thf(fact_5519_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space.simps(2)
thf(fact_5520_VEBT__internal_Ospace_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space @ X2 )
        = Y2 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y2
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space.elims
thf(fact_5521_upd__rule,axiom,
    ! [I: nat,Xs2: list_o,A: array_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( hoare_6478655245392655262rray_o @ ( snga_assn_o @ A @ Xs2 ) @ ( array_upd_o @ I @ X2 @ A )
        @ ^ [R: array_o] : ( times_times_assn @ ( snga_assn_o @ A @ ( list_update_o @ Xs2 @ I @ X2 ) ) @ ( pure_assn @ ( R = A ) ) ) ) ) ).

% upd_rule
thf(fact_5522_upd__rule,axiom,
    ! [I: nat,Xs2: list_nat,A: array_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( hoare_6807272225193264096ay_nat @ ( snga_assn_nat @ A @ Xs2 ) @ ( array_upd_nat @ I @ X2 @ A )
        @ ^ [R: array_nat] : ( times_times_assn @ ( snga_assn_nat @ A @ ( list_update_nat @ Xs2 @ I @ X2 ) ) @ ( pure_assn @ ( R = A ) ) ) ) ) ).

% upd_rule
thf(fact_5523_upd__rule,axiom,
    ! [I: nat,Xs2: list_int,A: array_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( hoare_2629421205684067388ay_int @ ( snga_assn_int @ A @ Xs2 ) @ ( array_upd_int @ I @ X2 @ A )
        @ ^ [R: array_int] : ( times_times_assn @ ( snga_assn_int @ A @ ( list_update_int @ Xs2 @ I @ X2 ) ) @ ( pure_assn @ ( R = A ) ) ) ) ) ).

% upd_rule
thf(fact_5524_upd__rule,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,A: array_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( hoare_3353465787467722821_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_upd_VEBT_VEBTi @ I @ X2 @ A )
        @ ^ [R: array_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) ) @ ( pure_assn @ ( R = A ) ) ) ) ) ).

% upd_rule
thf(fact_5525_length__rule,axiom,
    ! [A: array_o,Xs2: list_o] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_o @ A @ Xs2 ) @ ( array_len_o @ A )
      @ ^ [R: nat] :
          ( times_times_assn @ ( snga_assn_o @ A @ Xs2 )
          @ ( pure_assn
            @ ( R
              = ( size_size_list_o @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_5526_length__rule,axiom,
    ! [A: array_nat,Xs2: list_nat] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A @ Xs2 ) @ ( array_len_nat @ A )
      @ ^ [R: nat] :
          ( times_times_assn @ ( snga_assn_nat @ A @ Xs2 )
          @ ( pure_assn
            @ ( R
              = ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_5527_length__rule,axiom,
    ! [A: array_int,Xs2: list_int] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_int @ A @ Xs2 ) @ ( array_len_int @ A )
      @ ^ [R: nat] :
          ( times_times_assn @ ( snga_assn_int @ A @ Xs2 )
          @ ( pure_assn
            @ ( R
              = ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_5528_length__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_len_VEBT_VEBTi @ A )
      @ ^ [R: nat] :
          ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 )
          @ ( pure_assn
            @ ( R
              = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_5529_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ 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_5530_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is2: list_real] :
      ( ( ( ex_ass463751140784270563_VEBTi @ ( snga_assn_VEBT_VEBTi @ Tree_array ) )
        = top_top_assn )
     => ( ( heap_Time_return_nat @ ( size_size_list_real @ Tree_is2 ) )
        = ( array_len_VEBT_VEBTi @ Tree_array ) ) ) ).

% length_corresp
thf(fact_5531_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is2: list_o] :
      ( ( ( ex_ass463751140784270563_VEBTi @ ( snga_assn_VEBT_VEBTi @ Tree_array ) )
        = top_top_assn )
     => ( ( heap_Time_return_nat @ ( size_size_list_o @ Tree_is2 ) )
        = ( array_len_VEBT_VEBTi @ Tree_array ) ) ) ).

% length_corresp
thf(fact_5532_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is2: list_nat] :
      ( ( ( ex_ass463751140784270563_VEBTi @ ( snga_assn_VEBT_VEBTi @ Tree_array ) )
        = top_top_assn )
     => ( ( heap_Time_return_nat @ ( size_size_list_nat @ Tree_is2 ) )
        = ( array_len_VEBT_VEBTi @ Tree_array ) ) ) ).

% length_corresp
thf(fact_5533_length__corresp,axiom,
    ! [Tree_array: array_VEBT_VEBTi,Tree_is2: list_int] :
      ( ( ( ex_ass463751140784270563_VEBTi @ ( snga_assn_VEBT_VEBTi @ Tree_array ) )
        = top_top_assn )
     => ( ( heap_Time_return_nat @ ( size_size_list_int @ Tree_is2 ) )
        = ( array_len_VEBT_VEBTi @ Tree_array ) ) ) ).

% length_corresp
thf(fact_5534_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_5535_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_5536_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_5537_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_5538_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_5539_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_5540_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_5541_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_5542_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_5543_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_5544_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_5545_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_5546_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_5547_vebt__memberi_Osimps,axiom,
    ( vEBT_vebt_memberi
    = ( ^ [T2: vEBT_VEBTi,X: nat] :
          ( vEBT_c6104975476656191286Heap_o
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeList3: array_VEBT_VEBTi,Summary3: 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 @ Deg4 @ 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                              @ ^ [H: nat] :
                                  ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [L2: nat] :
                                      ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList3 )
                                      @ ^ [Len: nat] :
                                          ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                          @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList3 @ H )
                                            @ ^ [Th: vEBT_VEBTi] : ( vEBT_vebt_memberi @ Th @ L2 ) )
                                          @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A4: $o,B4: $o] :
              ( heap_Time_return_o
              @ ( ( ( X = zero_zero_nat )
                 => A4 )
                & ( ( X != zero_zero_nat )
                 => ( ( ( X = one_one_nat )
                     => B4 )
                    & ( X = one_one_nat ) ) ) ) )
          @ T2 ) ) ) ).

% vebt_memberi.simps
thf(fact_5548_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_5549_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_5550_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_5551_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_5552_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_5553_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_5554_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_5555_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5556_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5557_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5558_case__prod__conv,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5559_case__prod__conv,axiom,
    ! [F: int > int > int,A: int,B: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5560_map__fst__mk__snd,axiom,
    ! [K: produc8923325533196201883nteger,L: list_P1316552470764441098e_term] :
      ( ( map_Pr5817245686071818838e_term @ produc7822682618958472924nteger
        @ ( map_Pr3119800356862521302nteger
          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term] : ( produc8603105652947943368nteger @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5561_map__fst__mk__snd,axiom,
    ! [K: produc3658429121746597890et_nat,L: list_P7985473006766602707_nat_o] :
      ( ( map_Pr2899365764813592969_nat_o @ produc995936583742144908et_nat
        @ ( map_Pr1470255794198741767et_nat
          @ ^ [X: produc3658429121746597890et_nat > $o] : ( produc5001842942810119800et_nat @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5562_map__fst__mk__snd,axiom,
    ! [K: produc3925858234332021118et_nat,L: list_P7985473006766602707_nat_o] :
      ( ( map_Pr2606866347116058893_nat_o @ produc180342877477747464et_nat
        @ ( map_Pr8307916803881336451et_nat
          @ ^ [X: produc3658429121746597890et_nat > $o] : ( produc2245416461498447860et_nat @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5563_map__fst__mk__snd,axiom,
    ! [K: product_prod_int_int,L: list_P1743416141875011707e_term] :
      ( ( map_Pr5987089435700421812e_term @ produc6816164490631068361nt_int
        @ ( map_Pr7573407676697679540nt_int
          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term] : ( produc5700946648718959541nt_int @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5564_map__fst__mk__snd,axiom,
    ! [K: product_prod_int_int,L: list_i8448526496819171953e_term] :
      ( ( map_Pr7726103239032798900e_term @ produc6230002227079971283nt_int
        @ ( map_in8886716077063074740nt_int
          @ ^ [X: int > option6357759511663192854e_term] : ( produc4305682042979456191nt_int @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5565_map__fst__mk__snd,axiom,
    ! [K: set_nat,L: list_h2745205591237434579t_unit] :
      ( ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat
        @ ( map_he4567745686773187723et_nat
          @ ^ [X: heap_e7401611519738050253t_unit] : ( produc7507926704131184380et_nat @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5566_map__fst__mk__snd,axiom,
    ! [K: assn,L: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn
        @ ( map_as2373307505041272643n_assn
          @ ^ [X: assn] : ( produc118845697133431529n_assn @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5567_map__fst__mk__snd,axiom,
    ! [K: uint32,L: list_uint32] :
      ( ( map_Pr2530660914518622561uint32 @ produc9004433772639906525uint32
        @ ( map_ui5747794299746474503uint32
          @ ^ [X: uint32] : ( produc1400373151660368625uint32 @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5568_map__fst__mk__snd,axiom,
    ! [K: nat,L: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5569_map__fst__mk__snd,axiom,
    ! [K: int,L: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int
        @ ( map_in7157766398909135175nt_int
          @ ^ [X: int] : ( product_Pair_int_int @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_5570_map__snd__mk__fst,axiom,
    ! [K: produc6241069584506657477e_term > option6357759511663192854e_term,L: list_P5578671422887162913nteger] :
      ( ( map_Pr3733713507888840127nteger @ produc7856867400915047194nteger @ ( map_Pr4561634935768196077nteger @ ( produc8603105652947943368nteger @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5571_map__snd__mk__fst,axiom,
    ! [K: produc3658429121746597890et_nat > $o,L: list_P9062070895058802706et_nat] :
      ( ( map_Pr3528469389347239238et_nat @ produc4011572625026189258et_nat @ ( map_Pr1906792144924755270et_nat @ ( produc5001842942810119800et_nat @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5572_map__snd__mk__fst,axiom,
    ! [K: produc3658429121746597890et_nat > $o,L: list_P2321686559999237006et_nat] :
      ( ( map_Pr4224684623976845886et_nat @ produc5374455773327741254et_nat @ ( map_Pr5173603106669912638et_nat @ ( produc2245416461498447860et_nat @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5573_map__snd__mk__fst,axiom,
    ! [K: produc8551481072490612790e_term > option6357759511663192854e_term,L: list_P5707943133018811711nt_int] :
      ( ( map_Pr8582622553364537520nt_int @ produc7328097813583171335nt_int @ ( map_Pr1898935522916328184nt_int @ ( produc5700946648718959541nt_int @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5574_map__snd__mk__fst,axiom,
    ! [K: int > option6357759511663192854e_term,L: list_P5707943133018811711nt_int] :
      ( ( map_Pr5958523780849169702nt_int @ produc3162348030201620241nt_int @ ( map_Pr1306541819098601986nt_int @ ( produc4305682042979456191nt_int @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5575_map__snd__mk__fst,axiom,
    ! [K: nat,L: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5576_map__snd__mk__fst,axiom,
    ! [K: heap_e7401611519738050253t_unit,L: list_set_nat] :
      ( ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat @ ( map_se5435525824004667726et_nat @ ( produc7507926704131184380et_nat @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5577_map__snd__mk__fst,axiom,
    ! [K: int,L: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ ( map_in7157766398909135175nt_int @ ( product_Pair_int_int @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5578_map__snd__mk__fst,axiom,
    ! [K: code_integer,L: list_Code_integer] :
      ( ( map_Pr1857711230949937460nteger @ produc6174133586879617921nteger @ ( map_Co3589949550033412536nteger @ ( produc1086072967326762835nteger @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5579_map__snd__mk__fst,axiom,
    ! [K: assn,L: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ ( map_as2373307505041272643n_assn @ ( produc118845697133431529n_assn @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_5580_top__option__def,axiom,
    ( top_top_option_assn
    = ( some_assn @ top_top_assn ) ) ).

% top_option_def
thf(fact_5581_top__option__def,axiom,
    ( top_to1083748111577038690t_real
    = ( some_set_real @ top_top_set_real ) ) ).

% top_option_def
thf(fact_5582_top__option__def,axiom,
    ( top_to4826455019444611206et_nat
    = ( some_set_nat @ top_top_set_nat ) ) ).

% top_option_def
thf(fact_5583_top__option__def,axiom,
    ( top_to6745749650031393671t_char
    = ( some_set_char @ top_top_set_char ) ) ).

% top_option_def
thf(fact_5584_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
        @ ^ [X15: nat,X25: nat] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5585_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X25: int] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5586_prod_Ocase__distrib,axiom,
    ! [H2: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X15: int,X25: int] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5587_prod_Ocase__distrib,axiom,
    ! [H2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X25: int] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5588_prod_Ocase__distrib,axiom,
    ! [H2: int > int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X15: int,X25: int] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5589_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
        @ ^ [X15: nat,X25: nat] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5590_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
        @ ^ [X15: nat,X25: nat] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5591_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X25: int] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5592_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X15: int,X25: int] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5593_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X15: int,X25: int] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5594_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat,X1: nat,X23: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X23 ) )
      = ( F @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_5595_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X23: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X23 ) )
      = ( F @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_5596_old_Oprod_Ocase,axiom,
    ! [F: int > int > product_prod_int_int,X1: int,X23: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X23 ) )
      = ( F @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_5597_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X23: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X23 ) )
      = ( F @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_5598_old_Oprod_Ocase,axiom,
    ! [F: int > int > int,X1: int,X23: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X23 ) )
      = ( F @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_5599_merge__true__star__ctx,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ top_top_assn @ ( times_times_assn @ top_top_assn @ P ) )
      = ( times_times_assn @ top_top_assn @ P ) ) ).

% merge_true_star_ctx
thf(fact_5600_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_5601_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_5602_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_5603_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_5604_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_5605_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_5606_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_5607_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_5608_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_5609_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_5610_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_5611_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_5612_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_5613_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_5614_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_5615_fn__fst__conv,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_fst_nat_nat @ X ) ) )
      = ( produc2626176000494625587at_nat
        @ ^ [A4: nat,Uu3: nat] : ( F @ A4 ) ) ) ).

% fn_fst_conv
thf(fact_5616_fn__fst__conv,axiom,
    ! [F: nat > $o] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_fst_nat_nat @ X ) ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [A4: nat,Uu3: nat] : ( F @ A4 ) ) ) ).

% fn_fst_conv
thf(fact_5617_fn__fst__conv,axiom,
    ! [F: int > product_prod_int_int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_fst_int_int @ X ) ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,Uu3: int] : ( F @ A4 ) ) ) ).

% fn_fst_conv
thf(fact_5618_fn__fst__conv,axiom,
    ! [F: int > $o] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_fst_int_int @ X ) ) )
      = ( produc4947309494688390418_int_o
        @ ^ [A4: int,Uu3: int] : ( F @ A4 ) ) ) ).

% fn_fst_conv
thf(fact_5619_fn__fst__conv,axiom,
    ! [F: int > int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_fst_int_int @ X ) ) )
      = ( produc8211389475949308722nt_int
        @ ^ [A4: int,Uu3: int] : ( F @ A4 ) ) ) ).

% fn_fst_conv
thf(fact_5620_fst__def,axiom,
    ( produc1824681642469235216et_nat
    = ( produc8289649279824296164t_unit
      @ ^ [X15: heap_e7401611519738050253t_unit,X25: set_nat] : X15 ) ) ).

% fst_def
thf(fact_5621_fst__def,axiom,
    ( produc9167289414957590229n_assn
    = ( produc2152611005075324454n_assn
      @ ^ [X15: assn,X25: assn] : X15 ) ) ).

% fst_def
thf(fact_5622_fst__def,axiom,
    ( produc9004433772639906525uint32
    = ( produc3183254182024347122uint32
      @ ^ [X15: uint32,X25: uint32] : X15 ) ) ).

% fst_def
thf(fact_5623_fst__def,axiom,
    ( product_fst_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X15: nat,X25: nat] : X15 ) ) ).

% fst_def
thf(fact_5624_fst__def,axiom,
    ( product_fst_int_int
    = ( produc8211389475949308722nt_int
      @ ^ [X15: int,X25: int] : X15 ) ) ).

% fst_def
thf(fact_5625_fn__snd__conv,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_snd_nat_nat @ X ) ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Uu3: nat] : F ) ) ).

% fn_snd_conv
thf(fact_5626_fn__snd__conv,axiom,
    ! [F: nat > $o] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_snd_nat_nat @ X ) ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [Uu3: nat] : F ) ) ).

% fn_snd_conv
thf(fact_5627_fn__snd__conv,axiom,
    ! [F: int > product_prod_int_int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_snd_int_int @ X ) ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Uu3: int] : F ) ) ).

% fn_snd_conv
thf(fact_5628_fn__snd__conv,axiom,
    ! [F: int > $o] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_snd_int_int @ X ) ) )
      = ( produc4947309494688390418_int_o
        @ ^ [Uu3: int] : F ) ) ).

% fn_snd_conv
thf(fact_5629_fn__snd__conv,axiom,
    ! [F: int > int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_snd_int_int @ X ) ) )
      = ( produc8211389475949308722nt_int
        @ ^ [Uu3: int] : F ) ) ).

% fn_snd_conv
thf(fact_5630_snd__def,axiom,
    ( product_snd_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X15: nat,X25: nat] : X25 ) ) ).

% snd_def
thf(fact_5631_snd__def,axiom,
    ( produc8586169260539613262et_nat
    = ( produc7138653640177796783et_nat
      @ ^ [X15: heap_e7401611519738050253t_unit,X25: set_nat] : X25 ) ) ).

% snd_def
thf(fact_5632_snd__def,axiom,
    ( produc6174133586879617921nteger
    = ( produc7469247390737344965nteger
      @ ^ [X15: code_integer,X25: code_integer] : X25 ) ) ).

% snd_def
thf(fact_5633_snd__def,axiom,
    ( produc2051961928117032727n_assn
    = ( produc2152611005075324454n_assn
      @ ^ [X15: assn,X25: assn] : X25 ) ) ).

% snd_def
thf(fact_5634_snd__def,axiom,
    ( product_snd_int_int
    = ( produc8211389475949308722nt_int
      @ ^ [X15: int,X25: int] : X25 ) ) ).

% snd_def
thf(fact_5635_pair__list__eqI,axiom,
    ! [Xs2: list_P3069071885182933823uint32,Ys: list_P3069071885182933823uint32] :
      ( ( ( map_Pr2530660914518622561uint32 @ produc9004433772639906525uint32 @ Xs2 )
        = ( map_Pr2530660914518622561uint32 @ produc9004433772639906525uint32 @ Ys ) )
     => ( ( ( map_Pr2530660914518622561uint32 @ produc1510406741064981791uint32 @ Xs2 )
          = ( map_Pr2530660914518622561uint32 @ produc1510406741064981791uint32 @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_5636_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_5637_pair__list__eqI,axiom,
    ! [Xs2: list_P9062070895058802706et_nat,Ys: list_P9062070895058802706et_nat] :
      ( ( ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat @ Xs2 )
        = ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat @ Ys ) )
     => ( ( ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat @ Xs2 )
          = ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_5638_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_5639_pair__list__eqI,axiom,
    ! [Xs2: list_P5578671422887162913nteger,Ys: list_P5578671422887162913nteger] :
      ( ( ( map_Pr1857711230949937460nteger @ produc8508995932063986495nteger @ Xs2 )
        = ( map_Pr1857711230949937460nteger @ produc8508995932063986495nteger @ Ys ) )
     => ( ( ( map_Pr1857711230949937460nteger @ produc6174133586879617921nteger @ Xs2 )
          = ( map_Pr1857711230949937460nteger @ produc6174133586879617921nteger @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_5640_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_5641_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_5642_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_5643_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_5644_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_5645_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_5646_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_5647_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_5648_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_5649_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_5650_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_5651_mod__star__trueE,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H5 ) ) ).

% mod_star_trueE
thf(fact_5652_mod__star__trueI,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H2 )
     => ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_5653_case__prod__rule,axiom,
    ! [X2: produc7773217078559923341nt_int,P: assn,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_Time_Heap_nat,Q: nat > assn] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( X2
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( hoare_3067605981109127869le_nat @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_3067605981109127869le_nat @ P @ ( produc5434833944018601328ap_nat @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5654_case__prod__rule,axiom,
    ! [X2: produc7773217078559923341nt_int,P: assn,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_Time_Heap_o,Q: $o > assn] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( X2
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( hoare_hoare_triple_o @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_hoare_triple_o @ P @ ( produc5569244097790701070Heap_o @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5655_case__prod__rule,axiom,
    ! [X2: produc7773217078559923341nt_int,P: assn,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( X2
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( hoare_1429296392585015714_VEBTi @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ P @ ( produc7863211024854704597_VEBTi @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5656_case__prod__rule,axiom,
    ! [X2: produc7773217078559923341nt_int,P: assn,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( X2
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( hoare_8945653483474564448t_unit @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_8945653483474564448t_unit @ P @ ( produc4138331393287349635t_unit @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5657_case__prod__rule,axiom,
    ! [X2: produc7773217078559923341nt_int,P: assn,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( X2
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( hoare_7629718768684598413on_nat @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_7629718768684598413on_nat @ P @ ( produc6304067735488162624on_nat @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5658_case__prod__rule,axiom,
    ! [X2: produc3925858234332021118et_nat,P: assn,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_Time_Heap_nat,Q: nat > assn] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( X2
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( hoare_3067605981109127869le_nat @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_3067605981109127869le_nat @ P @ ( produc4044197431756818551ap_nat @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5659_case__prod__rule,axiom,
    ! [X2: produc3925858234332021118et_nat,P: assn,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_Time_Heap_o,Q: $o > assn] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( X2
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( hoare_hoare_triple_o @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_hoare_triple_o @ P @ ( produc6721000200443865927Heap_o @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5660_case__prod__rule,axiom,
    ! [X2: produc3925858234332021118et_nat,P: assn,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( X2
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( hoare_1429296392585015714_VEBTi @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ P @ ( produc7013922102172525020_VEBTi @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5661_case__prod__rule,axiom,
    ! [X2: produc3925858234332021118et_nat,P: assn,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( X2
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( hoare_8945653483474564448t_unit @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_8945653483474564448t_unit @ P @ ( produc8461005105828026684t_unit @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5662_case__prod__rule,axiom,
    ! [X2: produc3925858234332021118et_nat,P: assn,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( X2
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( hoare_7629718768684598413on_nat @ P @ ( F @ A3 @ B2 ) @ Q ) )
     => ( hoare_7629718768684598413on_nat @ P @ ( produc4900045171585128391on_nat @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_5663_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_5664_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_5665_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_5666_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_5667_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_5668_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_5669_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_5670_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_5671_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_5672_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_5673_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_5674_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_5675_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_5676_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_5677_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_5678_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_5679_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_5680_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_5681_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_5682_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_5683_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_5684_mod__star__trueE_H,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( ( produc1824681642469235216et_nat @ H5 )
              = ( produc1824681642469235216et_nat @ H2 ) )
           => ( ( ord_less_eq_set_nat @ ( produc8586169260539613262et_nat @ H5 ) @ ( produc8586169260539613262et_nat @ H2 ) )
             => ~ ( rep_assn @ P @ H5 ) ) ) ) ).

% mod_star_trueE'
thf(fact_5685_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Leafi @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBTi.size_gen(2)
thf(fact_5686_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( 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 @ TreeList ) @ zero_zero_real ) ) ) ).

% VEBT_internal.cnt.simps(2)
thf(fact_5687_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: real] :
      ( ( ( vEBT_VEBT_cnt @ X2 )
        = Y2 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y2 != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) ) ) ) ) ).

% VEBT_internal.cnt.elims
thf(fact_5688_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T: vEBT_VEBT] : ( refine_Imp_refines_o @ ( vEBT_vebt_memberi @ Ti @ X2 ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X2 ) ) ).

% vebt_memberi_refines
thf(fact_5689_of__nat__code__if,axiom,
    ( semiri2565882477558803405uint32
    = ( ^ [N: nat] :
          ( if_uint32 @ ( N = zero_zero_nat ) @ zero_zero_uint32
          @ ( produc2417093276151063866uint32
            @ ^ [M7: nat,Q5: nat] : ( if_uint32 @ ( Q5 = zero_zero_nat ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( semiri2565882477558803405uint32 @ M7 ) ) @ ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( semiri2565882477558803405uint32 @ M7 ) ) @ one_one_uint32 ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5690_of__nat__code__if,axiom,
    ( semiri8819519690708144855l_num1
    = ( ^ [N: nat] :
          ( if_wor5778924947035936048l_num1 @ ( N = zero_zero_nat ) @ zero_z3563351764282998399l_num1
          @ ( produc6192303373133366212l_num1
            @ ^ [M7: nat,Q5: nat] : ( if_wor5778924947035936048l_num1 @ ( Q5 = zero_zero_nat ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( semiri8819519690708144855l_num1 @ M7 ) ) @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( semiri8819519690708144855l_num1 @ M7 ) ) @ one_on7727431528512463931l_num1 ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5691_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M7: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M7 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M7 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5692_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M7: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M7 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M7 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5693_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M7: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M7 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M7 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5694_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M7: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M7 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M7 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5695_of__nat__code__if,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N: nat] :
          ( if_Code_integer @ ( N = zero_zero_nat ) @ zero_z3403309356797280102nteger
          @ ( produc1830744345554046123nteger
            @ ^ [M7: nat,Q5: nat] : ( if_Code_integer @ ( Q5 = zero_zero_nat ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M7 ) ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M7 ) ) @ one_one_Code_integer ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5696_vebt__succi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_succi @ Ti @ X2 ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X2 ) ) ).

% vebt_succi_refines
thf(fact_5697_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_5698_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_5699_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_5700_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_5701_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_5702_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_5703_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_5704_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_5705_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T2636463487746394924on_nat,F13: $o > $o > heap_T2636463487746394924on_nat,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F24: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B2: $o] : ( refine7594492741263601813on_nat @ ( F1 @ A3 @ B2 ) @ ( F13 @ A3 @ B2 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine7594492741263601813on_nat @ ( F22 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( F24 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
         => ( refine7594492741263601813on_nat @ ( vEBT_c6250501799366334488on_nat @ F22 @ F1 @ Ti ) @ ( vEBT_c6250501799366334488on_nat @ F24 @ F13 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5706_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T8145700208782473153_VEBTi,F13: $o > $o > heap_T8145700208782473153_VEBTi,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F24: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B2: $o] : ( refine5565527176597971370_VEBTi @ ( F1 @ A3 @ B2 ) @ ( F13 @ A3 @ B2 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F22 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( F24 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
         => ( refine5565527176597971370_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F22 @ F1 @ Ti ) @ ( vEBT_c6028912655521741485_VEBTi @ F24 @ F13 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5707_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_Time_Heap_o,F13: $o > $o > heap_Time_Heap_o,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F24: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B2: $o] : ( refine_Imp_refines_o @ ( F1 @ A3 @ B2 ) @ ( F13 @ A3 @ B2 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine_Imp_refines_o @ ( F22 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( F24 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
         => ( refine_Imp_refines_o @ ( vEBT_c6104975476656191286Heap_o @ F22 @ F1 @ Ti ) @ ( vEBT_c6104975476656191286Heap_o @ F24 @ F13 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5708_case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_5709_case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc1437786849005270451_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_5710_case__prodI2,axiom,
    ! [P4: produc2732055786443039994et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
          ( ( P4
            = ( produc2245416461498447860et_nat @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc838355143741117751_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_5711_case__prodI2,axiom,
    ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc5700946648718959541nt_int @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_5712_case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_5713_case__prodI2,axiom,
    ! [P4: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B2: nat] :
          ( ( P4
            = ( product_Pair_nat_nat @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc6081775807080527818_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_5714_case__prodI2,axiom,
    ! [P4: product_prod_int_int,C: int > int > $o] :
      ( ! [A3: int,B2: int] :
          ( ( P4
            = ( product_Pair_int_int @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_5715_case__prodI,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% case_prodI
thf(fact_5716_case__prodI,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( F @ A @ B )
     => ( produc1437786849005270451_nat_o @ F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_5717_case__prodI,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( F @ A @ B )
     => ( produc838355143741117751_nat_o @ F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_5718_case__prodI,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_5719_case__prodI,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_5720_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_5721_case__prodI,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( F @ A @ B )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_5722_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5723_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: vEBT_VEBT,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_VEBT_VEBT] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member_VEBT_VEBT @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_VEBT_VEBT @ Z @ ( produc6639173862353101480T_VEBT @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5724_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5725_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5726_mem__case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5727_mem__case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,Z: vEBT_VEBT,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_VEBT_VEBT] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( member_VEBT_VEBT @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_VEBT_VEBT @ Z @ ( produc5559244600637454639T_VEBT @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5728_mem__case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,Z: real,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_real] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_real @ Z @ ( produc7349554562891328263t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5729_mem__case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5730_mem__case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5731_mem__case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,Z: vEBT_VEBT,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_VEBT_VEBT] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( member_VEBT_VEBT @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_VEBT_VEBT @ Z @ ( produc385966054578180063T_VEBT @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_5732_mem__case__prodI,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5733_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_VEBT_VEBT,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_VEBT_VEBT @ Z @ ( C @ A @ B ) )
     => ( member_VEBT_VEBT @ Z @ ( produc6639173862353101480T_VEBT @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5734_mem__case__prodI,axiom,
    ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5735_mem__case__prodI,axiom,
    ! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5736_mem__case__prodI,axiom,
    ! [Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5737_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_VEBT_VEBT,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_VEBT_VEBT @ Z @ ( C @ A @ B ) )
     => ( member_VEBT_VEBT @ Z @ ( produc5559244600637454639T_VEBT @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5738_mem__case__prodI,axiom,
    ! [Z: real,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_real,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc7349554562891328263t_real @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5739_mem__case__prodI,axiom,
    ! [Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5740_mem__case__prodI,axiom,
    ! [Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5741_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_VEBT_VEBT,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_VEBT_VEBT @ Z @ ( C @ A @ B ) )
     => ( member_VEBT_VEBT @ Z @ ( produc385966054578180063T_VEBT @ C @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5742_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_complex
            @ ^ [S4: complex] : P )
          = top_top_set_complex ) )
      & ( ~ P
       => ( ( collect_complex
            @ ^ [S4: complex] : P )
          = bot_bot_set_complex ) ) ) ).

% Collect_const
thf(fact_5743_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec213857154873943460nt_int
            @ ^ [S4: product_prod_int_int] : P )
          = top_to4366644338036079209nt_int ) )
      & ( ~ P
       => ( ( collec213857154873943460nt_int
            @ ^ [S4: product_prod_int_int] : P )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_const
thf(fact_5744_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_int
            @ ^ [S4: int] : P )
          = top_top_set_int ) )
      & ( ~ P
       => ( ( collect_int
            @ ^ [S4: int] : P )
          = bot_bot_set_int ) ) ) ).

% Collect_const
thf(fact_5745_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_real
            @ ^ [S4: real] : P )
          = top_top_set_real ) )
      & ( ~ P
       => ( ( collect_real
            @ ^ [S4: real] : P )
          = bot_bot_set_real ) ) ) ).

% Collect_const
thf(fact_5746_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S4: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S4: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_5747_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_char
            @ ^ [S4: char] : P )
          = top_top_set_char ) )
      & ( ~ P
       => ( ( collect_char
            @ ^ [S4: char] : P )
          = bot_bot_set_char ) ) ) ).

% Collect_const
thf(fact_5748_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A4: nat,B4: nat] : P ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A4: nat,B4: nat] : P ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_5749_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec213857154873943460nt_int
            @ ( produc4947309494688390418_int_o
              @ ^ [A4: int,B4: int] : P ) )
          = top_to4366644338036079209nt_int ) )
      & ( ~ P
       => ( ( collec213857154873943460nt_int
            @ ( produc4947309494688390418_int_o
              @ ^ [A4: int,B4: int] : P ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_const_case_prod
thf(fact_5750_Diff__UNIV,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ top_top_set_int )
      = bot_bot_set_int ) ).

% Diff_UNIV
thf(fact_5751_Diff__UNIV,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ top_top_set_real )
      = bot_bot_set_real ) ).

% Diff_UNIV
thf(fact_5752_Diff__UNIV,axiom,
    ! [A2: set_char] :
      ( ( minus_minus_set_char @ A2 @ top_top_set_char )
      = bot_bot_set_char ) ).

% Diff_UNIV
thf(fact_5753_Diff__UNIV,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ top_top_set_nat )
      = bot_bot_set_nat ) ).

% Diff_UNIV
thf(fact_5754_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_5755_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_5756_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_5757_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_5758_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_5759_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_5760_mem__case__prodE,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
      ( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5761_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_VEBT_VEBT,P4: produc7773217078559923341nt_int] :
      ( ( member_VEBT_VEBT @ Z @ ( produc6639173862353101480T_VEBT @ C @ P4 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_VEBT_VEBT @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5762_mem__case__prodE,axiom,
    ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
      ( ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5763_mem__case__prodE,axiom,
    ! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,P4: produc7773217078559923341nt_int] :
      ( ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P4 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5764_mem__case__prodE,axiom,
    ! [Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat,P4: produc3925858234332021118et_nat] :
      ( ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ P4 ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5765_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_VEBT_VEBT,P4: produc3925858234332021118et_nat] :
      ( ( member_VEBT_VEBT @ Z @ ( produc5559244600637454639T_VEBT @ C @ P4 ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( member_VEBT_VEBT @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5766_mem__case__prodE,axiom,
    ! [Z: real,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_real,P4: produc3925858234332021118et_nat] :
      ( ( member_real @ Z @ ( produc7349554562891328263t_real @ C @ P4 ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5767_mem__case__prodE,axiom,
    ! [Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int,P4: produc3925858234332021118et_nat] :
      ( ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ P4 ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5768_mem__case__prodE,axiom,
    ! [Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc1908205239877642774nteger] :
      ( ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ P4 ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5769_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_VEBT_VEBT,P4: produc1908205239877642774nteger] :
      ( ( member_VEBT_VEBT @ Z @ ( produc385966054578180063T_VEBT @ C @ P4 ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X3 @ Y3 ) )
           => ~ ( member_VEBT_VEBT @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5770_subset__UNIV,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).

% subset_UNIV
thf(fact_5771_subset__UNIV,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_5772_subset__UNIV,axiom,
    ! [A2: set_char] : ( ord_less_eq_set_char @ A2 @ top_top_set_char ) ).

% subset_UNIV
thf(fact_5773_subset__UNIV,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).

% subset_UNIV
thf(fact_5774_case__prodE,axiom,
    ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
      ( ( produc6253627499356882019eger_o @ C @ P4 )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5775_case__prodE,axiom,
    ! [C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P4: produc3925858234332021118et_nat] :
      ( ( produc1437786849005270451_nat_o @ C @ P4 )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5776_case__prodE,axiom,
    ! [C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,P4: produc2732055786443039994et_nat] :
      ( ( produc838355143741117751_nat_o @ C @ P4 )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
            ( ( P4
              = ( produc2245416461498447860et_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5777_case__prodE,axiom,
    ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
      ( ( produc1573362020775583542_int_o @ C @ P4 )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P4
              = ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5778_case__prodE,axiom,
    ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
      ( ( produc2558449545302689196_int_o @ C @ P4 )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5779_case__prodE,axiom,
    ! [C: nat > nat > $o,P4: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P4 )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5780_case__prodE,axiom,
    ! [C: int > int > $o,P4: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P4 )
     => ~ ! [X3: int,Y3: int] :
            ( ( P4
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5781_case__prodD,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5782_case__prodD,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc1437786849005270451_nat_o @ F @ ( produc5001842942810119800et_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5783_case__prodD,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc838355143741117751_nat_o @ F @ ( produc2245416461498447860et_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5784_case__prodD,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5785_case__prodD,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5786_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5787_case__prodD,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5788_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,C5: heap_T2636463487746394924on_nat] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ( refine7594492741263601813on_nat @ C5 @ C )
       => ( hoare_7629718768684598413on_nat @ P @ C5 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_5789_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,C5: heap_Time_Heap_nat] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ( refine1365783493865988805es_nat @ C5 @ C )
       => ( hoare_3067605981109127869le_nat @ P @ C5 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_5790_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,C5: heap_Time_Heap_o] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ( refine_Imp_refines_o @ C5 @ C )
       => ( hoare_hoare_triple_o @ P @ C5 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_5791_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,C5: heap_T8145700208782473153_VEBTi] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ( refine5565527176597971370_VEBTi @ C5 @ C )
       => ( hoare_1429296392585015714_VEBTi @ P @ C5 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_5792_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn,C5: heap_T5738788834812785303t_unit] :
      ( ( hoare_8945653483474564448t_unit @ P @ C @ Q )
     => ( ( refine451278484176762712t_unit @ C5 @ C )
       => ( hoare_8945653483474564448t_unit @ P @ C5 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_5793_htt__refine,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,T: nat,C5: heap_Time_Heap_nat] :
      ( ( time_htt_nat @ P @ C @ Q @ T )
     => ( ( refine1365783493865988805es_nat @ C5 @ C )
       => ( time_htt_nat @ P @ C5 @ Q @ T ) ) ) ).

% htt_refine
thf(fact_5794_htt__refine,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,T: nat,C5: heap_Time_Heap_o] :
      ( ( time_htt_o @ P @ C @ Q @ T )
     => ( ( refine_Imp_refines_o @ C5 @ C )
       => ( time_htt_o @ P @ C5 @ Q @ T ) ) ) ).

% htt_refine
thf(fact_5795_htt__refine,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat,C5: heap_T2636463487746394924on_nat] :
      ( ( time_htt_option_nat @ P @ C @ Q @ T )
     => ( ( refine7594492741263601813on_nat @ C5 @ C )
       => ( time_htt_option_nat @ P @ C5 @ Q @ T ) ) ) ).

% htt_refine
thf(fact_5796_htt__refine,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat,C5: heap_T8145700208782473153_VEBTi] :
      ( ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T )
     => ( ( refine5565527176597971370_VEBTi @ C5 @ C )
       => ( time_htt_VEBT_VEBTi @ P @ C5 @ Q @ T ) ) ) ).

% htt_refine
thf(fact_5797_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_5798_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_5799_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_5800_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_5801_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_5802_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_5803_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_5804_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_5805_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M6: heap_Time_Heap_o,F: $o > heap_T2636463487746394924on_nat,F3: $o > heap_T2636463487746394924on_nat] :
      ( ( refine_Imp_refines_o @ M @ M6 )
     => ( ! [X3: $o] : ( refine7594492741263601813on_nat @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine7594492741263601813on_nat @ ( heap_T6306279297776390513on_nat @ M @ F ) @ ( heap_T6306279297776390513on_nat @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5806_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_5807_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc827990862158126777uint32,A2: uint32 > uint32 > $o] :
      ( ( member8027108493173000802uint32 @ X2 @ ( collec104543260748735396uint32 @ ( produc6247642326928354578nt32_o @ A2 ) ) )
     => ( A2 @ ( produc9004433772639906525uint32 @ X2 ) @ ( produc1510406741064981791uint32 @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_5808_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit > set_nat > $o] :
      ( ( member6260224972018164377et_nat @ X2 @ ( collec6777745107090321879et_nat @ ( produc6968281743908733999_nat_o @ A2 ) ) )
     => ( A2 @ ( produc1824681642469235216et_nat @ X2 ) @ ( produc8586169260539613262et_nat @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_5809_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc8923325533196201883nteger,A2: code_integer > code_integer > $o] :
      ( ( member157494554546826820nteger @ X2 @ ( collec7766031446232907142nteger @ ( produc2066375834425727024eger_o @ A2 ) ) )
     => ( A2 @ ( produc8508995932063986495nteger @ X2 ) @ ( produc6174133586879617921nteger @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_5810_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc6575502325842934193n_assn,A2: assn > assn > $o] :
      ( ( member7957490590177025114n_assn @ X2 @ ( collec1604292450182575004n_assn @ ( produc7274209992780475162assn_o @ A2 ) ) )
     => ( A2 @ ( produc9167289414957590229n_assn @ X2 ) @ ( produc2051961928117032727n_assn @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_5811_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: product_prod_nat_nat,A2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A2 ) ) )
     => ( A2 @ ( product_fst_nat_nat @ X2 ) @ ( product_snd_nat_nat @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_5812_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: product_prod_int_int,A2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ X2 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) )
     => ( A2 @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_5813_refines__option,axiom,
    ! [A: 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] :
      ( ( A = 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 @ A ) @ ( case_o1442776274061689234at_nat @ M12 @ M23 @ A5 ) ) ) ) ) ).

% refines_option
thf(fact_5814_not__UNIV__le__Icc,axiom,
    ! [L: nat,H2: nat] :
      ~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) ) ).

% not_UNIV_le_Icc
thf(fact_5815_not__UNIV__le__Icc,axiom,
    ! [L: int,H2: int] :
      ~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) ) ).

% not_UNIV_le_Icc
thf(fact_5816_not__UNIV__le__Icc,axiom,
    ! [L: real,H2: real] :
      ~ ( ord_less_eq_set_real @ top_top_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) ) ).

% not_UNIV_le_Icc
thf(fact_5817_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_5818_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M7: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M7 @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M7 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M7 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_5819_ceiling__ge__round,axiom,
    ! [X2: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).

% ceiling_ge_round
thf(fact_5820_refines__assert_H__bind,axiom,
    ! [P4: heap_Time_Heap_o,Q2: heap_Time_Heap_o,Phi: $o] :
      ( ( refine_Imp_refines_o @ P4 @ Q2 )
     => ( refine_Imp_refines_o @ P4
        @ ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ Phi )
          @ ^ [Uu3: product_unit] : Q2 ) ) ) ).

% refines_assert'_bind
thf(fact_5821_refines__assert_H__bind,axiom,
    ! [P4: heap_T2636463487746394924on_nat,Q2: heap_T2636463487746394924on_nat,Phi: $o] :
      ( ( refine7594492741263601813on_nat @ P4 @ Q2 )
     => ( refine7594492741263601813on_nat @ P4
        @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ Phi )
          @ ^ [Uu3: product_unit] : Q2 ) ) ) ).

% refines_assert'_bind
thf(fact_5822_refines__assert_H__bind,axiom,
    ! [P4: heap_T8145700208782473153_VEBTi,Q2: heap_T8145700208782473153_VEBTi,Phi: $o] :
      ( ( refine5565527176597971370_VEBTi @ P4 @ Q2 )
     => ( refine5565527176597971370_VEBTi @ P4
        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ Phi )
          @ ^ [Uu3: product_unit] : Q2 ) ) ) ).

% refines_assert'_bind
thf(fact_5823_TBOUND__prod__case,axiom,
    ! [T: produc7773217078559923341nt_int,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_Time_Heap_o,Bnd: ( int > option6357759511663192854e_term ) > product_prod_int_int > nat] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( T
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( time_TBOUND_o @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_TBOUND_o @ ( produc5569244097790701070Heap_o @ F @ T ) @ ( produc6909675604869682876nt_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5824_TBOUND__prod__case,axiom,
    ! [T: produc7773217078559923341nt_int,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_Time_Heap_nat,Bnd: ( int > option6357759511663192854e_term ) > product_prod_int_int > nat] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( T
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( time_TBOUND_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_TBOUND_nat @ ( produc5434833944018601328ap_nat @ F @ T ) @ ( produc6909675604869682876nt_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5825_TBOUND__prod__case,axiom,
    ! [T: produc7773217078559923341nt_int,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_T8145700208782473153_VEBTi,Bnd: ( int > option6357759511663192854e_term ) > product_prod_int_int > nat] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( T
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_T5737551269749752165_VEBTi @ ( produc7863211024854704597_VEBTi @ F @ T ) @ ( produc6909675604869682876nt_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5826_TBOUND__prod__case,axiom,
    ! [T: produc7773217078559923341nt_int,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > heap_T2636463487746394924on_nat,Bnd: ( int > option6357759511663192854e_term ) > product_prod_int_int > nat] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( T
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( time_T8353473612707095248on_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_T8353473612707095248on_nat @ ( produc6304067735488162624on_nat @ F @ T ) @ ( produc6909675604869682876nt_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5827_TBOUND__prod__case,axiom,
    ! [T: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_Time_Heap_o,Bnd: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( T
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( time_TBOUND_o @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_TBOUND_o @ ( produc6721000200443865927Heap_o @ F @ T ) @ ( produc4286856396916569717at_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5828_TBOUND__prod__case,axiom,
    ! [T: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_Time_Heap_nat,Bnd: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( T
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( time_TBOUND_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_TBOUND_nat @ ( produc4044197431756818551ap_nat @ F @ T ) @ ( produc4286856396916569717at_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5829_TBOUND__prod__case,axiom,
    ! [T: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_T8145700208782473153_VEBTi,Bnd: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( T
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_T5737551269749752165_VEBTi @ ( produc7013922102172525020_VEBTi @ F @ T ) @ ( produc4286856396916569717at_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5830_TBOUND__prod__case,axiom,
    ! [T: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > heap_T2636463487746394924on_nat,Bnd: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( T
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( time_T8353473612707095248on_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_T8353473612707095248on_nat @ ( produc4900045171585128391on_nat @ F @ T ) @ ( produc4286856396916569717at_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5831_TBOUND__prod__case,axiom,
    ! [T: produc1908205239877642774nteger,F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > heap_Time_Heap_o,Bnd: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > nat] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( T
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( time_TBOUND_o @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_TBOUND_o @ ( produc7687182067898032279Heap_o @ F @ T ) @ ( produc1060995028572156869er_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5832_TBOUND__prod__case,axiom,
    ! [T: produc2285326912895808259nt_int,F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > heap_Time_Heap_o,Bnd: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > nat] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( T
            = ( produc5700946648718959541nt_int @ A3 @ B2 ) )
         => ( time_TBOUND_o @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( time_TBOUND_o @ ( produc5242196661281123332Heap_o @ F @ T ) @ ( produc4409929891536499634nt_nat @ Bnd @ T ) ) ) ).

% TBOUND_prod_case
thf(fact_5833_rel__of__def,axiom,
    ( rel_of7835217753297429671nteger
    = ( ^ [M7: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > option2651255830984564193nteger,P3: produc1908205239877642774nteger > $o] :
          ( collec6950949345391930603nteger
          @ ( produc6253627499356882019eger_o
            @ ^ [K4: produc6241069584506657477e_term > option6357759511663192854e_term,V4: produc8923325533196201883nteger] :
                ( ( ( M7 @ K4 )
                  = ( some_P6772290148444788224nteger @ V4 ) )
                & ( P3 @ ( produc8603105652947943368nteger @ K4 @ V4 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_5834_rel__of__def,axiom,
    ( rel_of4838799251197538391et_nat
    = ( ^ [M7: ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat,P3: produc3925858234332021118et_nat > $o] :
          ( collec1402215087704437587et_nat
          @ ( produc1437786849005270451_nat_o
            @ ^ [K4: produc3658429121746597890et_nat > $o,V4: produc3658429121746597890et_nat] :
                ( ( ( M7 @ K4 )
                  = ( some_P624177172695371229et_nat @ V4 ) )
                & ( P3 @ ( produc5001842942810119800et_nat @ K4 @ V4 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_5835_rel__of__def,axiom,
    ( rel_of7774016450764239315et_nat
    = ( ^ [M7: ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat,P3: produc2732055786443039994et_nat > $o] :
          ( collec5543584681430388431et_nat
          @ ( produc838355143741117751_nat_o
            @ ^ [K4: produc3658429121746597890et_nat > $o,V4: produc3925858234332021118et_nat] :
                ( ( ( M7 @ K4 )
                  = ( some_P750831030444334937et_nat @ V4 ) )
                & ( P3 @ ( produc2245416461498447860et_nat @ K4 @ V4 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_5836_rel__of__def,axiom,
    ( rel_of8306664904814525588nt_int
    = ( ^ [M7: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > option4624381673175914239nt_int,P3: produc2285326912895808259nt_int > $o] :
          ( collec1790188477890212312nt_int
          @ ( produc1573362020775583542_int_o
            @ ^ [K4: produc8551481072490612790e_term > option6357759511663192854e_term,V4: product_prod_int_int] :
                ( ( ( M7 @ K4 )
                  = ( some_P4184893108420464158nt_int @ V4 ) )
                & ( P3 @ ( produc5700946648718959541nt_int @ K4 @ V4 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_5837_rel__of__def,axiom,
    ( rel_of5543720577181062686nt_int
    = ( ^ [M7: ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int,P3: produc7773217078559923341nt_int > $o] :
          ( collec506566255779805410nt_int
          @ ( produc2558449545302689196_int_o
            @ ^ [K4: int > option6357759511663192854e_term,V4: product_prod_int_int] :
                ( ( ( M7 @ K4 )
                  = ( some_P4184893108420464158nt_int @ V4 ) )
                & ( P3 @ ( produc4305682042979456191nt_int @ K4 @ V4 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_5838_rel__of__def,axiom,
    ( rel_of_nat_nat
    = ( ^ [M7: nat > option_nat,P3: product_prod_nat_nat > $o] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [K4: nat,V4: nat] :
                ( ( ( M7 @ K4 )
                  = ( some_nat @ V4 ) )
                & ( P3 @ ( product_Pair_nat_nat @ K4 @ V4 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_5839_rel__of__def,axiom,
    ( rel_of_int_int
    = ( ^ [M7: int > option_int,P3: product_prod_int_int > $o] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [K4: int,V4: int] :
                ( ( ( M7 @ K4 )
                  = ( some_int @ V4 ) )
                & ( P3 @ ( product_Pair_int_int @ K4 @ V4 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_5840_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M7: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M7 @ N ) @ ( modulo_modulo_nat @ M7 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_5841_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_5842_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_5843_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_def
thf(fact_5844_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_def
thf(fact_5845_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_def
thf(fact_5846_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_5847_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_5848_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_5849_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_5850_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_5851_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_5852_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_5853_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_5854_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_5855_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_5856_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_5857_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_5858_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,H4: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N2: nat] :
      ( ! [Vebt_inserti: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: vEBT_VEBTi,N8: nat] :
              ( ( heap_T2071195472996403633_VEBTi @ ( Vebt_inserti @ A7 @ B7 ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa3: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N4: nat] :
              ( ( heap_T2071195472996403633_VEBTi
                @ ( vEBT_c6028912655521741485_VEBTi
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                      @ ^ [Minma: product_prod_nat_nat] :
                          ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                          @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [L2: nat] :
                                                ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                @ ^ [H: nat] :
                                                    ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                    @ ^ [Len: nat] :
                                                        ( if_Hea8453224502484754311_VEBTi
                                                        @ ( ( ord_less_nat @ H @ Len )
                                                          & ~ ( ( X3 = Mi3 )
                                                              | ( X3 = Ma3 ) ) )
                                                        @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                          @ ^ [Node: vEBT_VEBTi] :
                                                              ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                              @ ^ [Empt: $o] :
                                                                  ( heap_T1006145433769338483_VEBTi @ ( Vebt_inserti @ Node @ L2 )
                                                                  @ ^ [Newnode2: vEBT_VEBTi] :
                                                                      ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                      @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                          ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( Vebt_inserti @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                          @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                        @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A4: $o,B4: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X3 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B4 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X3 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
                  @ T4 )
                @ Ta
                @ Xa3
                @ R5
                @ N4 )
             => ( P @ T4 @ X3 @ Ta @ Xa3 @ R5 @ N4 ) ) )
     => ( ( heap_T2071195472996403633_VEBTi @ ( produc3255295512018472142_VEBTi @ vEBT_vebt_inserti @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc4924893227731358948_nat_o @ P @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% vebt_inserti.raw_induct
thf(fact_5859_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,H4: heap_e7401611519738050253t_unit,R3: option_nat,N2: nat] :
      ( ! [Vebt_predi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: option_nat,N8: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_predi @ A7 @ B7 ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa3: heap_e7401611519738050253t_unit,R5: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ^ [Mima2: product_prod_nat_nat] :
                          ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                          @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                            @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                              @ ^ [L2: nat] :
                                  ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [H: nat] :
                                      ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Minlow ) )
                                              @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi @ Aktnode @ L2 )
                                                @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                              @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi @ Summary3 @ H )
                                                @ ^ [Predsum: option_nat] :
                                                    ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                    @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X3 = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                  @ T4 )
                @ Ta
                @ Xa3
                @ R5
                @ N4 )
             => ( P @ T4 @ X3 @ Ta @ Xa3 @ R5 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc8911080112929139129on_nat @ vEBT_vebt_predi @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc6438938002899911481_nat_o @ P @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% vebt_predi.raw_induct
thf(fact_5860_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,H4: heap_e7401611519738050253t_unit,R3: option_nat,N2: nat] :
      ( ! [Vebt_succi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: option_nat,N8: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_succi @ A7 @ B7 ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa3: heap_e7401611519738050253t_unit,R5: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ^ [Mima2: product_prod_nat_nat] :
                          ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                          @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X3 @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                              @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                @ ^ [L2: nat] :
                                    ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [H: nat] :
                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Maxlow ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi @ Aktnode @ L2 )
                                                  @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi @ Summary3 @ H )
                                                  @ ^ [Succsum: option_nat] :
                                                      ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                      @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( X3 = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                  @ T4 )
                @ Ta
                @ Xa3
                @ R5
                @ N4 )
             => ( P @ T4 @ X3 @ Ta @ Xa3 @ R5 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc8911080112929139129on_nat @ vEBT_vebt_succi @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc6438938002899911481_nat_o @ P @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% vebt_succi.raw_induct
thf(fact_5861_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,H4: heap_e7401611519738050253t_unit,R3: $o,N2: nat] :
      ( ! [Vebt_memberi: vEBT_VEBTi > nat > heap_Time_Heap_o] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: $o,N8: nat] :
              ( ( heap_Time_effect_o @ ( Vebt_memberi @ A7 @ B7 ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa3: heap_e7401611519738050253t_unit,R5: $o,N4: nat] :
              ( ( heap_Time_effect_o
                @ ( vEBT_c6104975476656191286Heap_o
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeList3: array_VEBT_VEBTi,Summary3: 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 @ Deg4 @ 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [H: nat] :
                                          ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                          @ ^ [L2: nat] :
                                              ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList3 )
                                              @ ^ [Len: nat] :
                                                  ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                  @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList3 @ H )
                                                    @ ^ [Th: vEBT_VEBTi] : ( Vebt_memberi @ Th @ L2 ) )
                                                  @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A4: $o,B4: $o] :
                      ( heap_Time_return_o
                      @ ( ( ( X3 = zero_zero_nat )
                         => A4 )
                        & ( ( X3 != zero_zero_nat )
                         => ( ( ( X3 = one_one_nat )
                             => B4 )
                            & ( X3 = one_one_nat ) ) ) ) )
                  @ T4 )
                @ Ta
                @ Xa3
                @ R5
                @ N4 )
             => ( P @ T4 @ X3 @ Ta @ Xa3 @ R5 @ N4 ) ) )
     => ( ( heap_Time_effect_o @ ( produc770043135277712853Heap_o @ vEBT_vebt_memberi @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc1840203461219862933_nat_o @ P @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% vebt_memberi.raw_induct
thf(fact_5862_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_5863_VEBT__internal_Ovebt__succi_H_Osimps,axiom,
    ( vEBT_VEBT_vebt_succi
    = ( ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
              @ ^ [Uu3: product_unit] :
                  ( produc2190226783763740553on_nat
                  @ ^ [Info4: option4927543243414619207at_nat] :
                      ( produc2606485630176857543on_nat
                      @ ^ [Deg5: nat] :
                          ( produc6124225815318652422on_nat
                          @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                              ( heap_T5999496708990702694on_nat
                              @ ( refine_Imp_assert
                                @ ( ( Info4 = Info3 )
                                  & ( Deg5 = Deg4 )
                                  & ( vEBT_is_Node @ T2 ) ) )
                              @ ^ [Uv3: product_unit] :
                                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                  @ ^ [Mima2: product_prod_nat_nat] :
                                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                                        @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                          @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [L2: nat] :
                                                ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                @ ^ [H: nat] :
                                                    ( heap_T5999496708990702694on_nat
                                                    @ ( refine_Imp_assert
                                                      @ ( L2
                                                        = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    @ ^ [Uw3: product_unit] :
                                                        ( heap_T5999496708990702694on_nat
                                                        @ ( refine_Imp_assert
                                                          @ ( H
                                                            = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                        @ ^ [Ux3: product_unit] :
                                                            ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                            @ ^ [Uy3: product_unit] :
                                                                ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                        @ ^ [Uz3: product_unit] :
                                                                            ( if_Hea5867803462524415986on_nat
                                                                            @ ( ( Maxlow != none_nat )
                                                                              & ( vEBT_VEBT_less @ ( some_nat @ L2 ) @ Maxlow ) )
                                                                            @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_succi @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                              @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                            @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_succi @ Summary4 @ Summary3 @ H )
                                                                              @ ^ [Succsum: option_nat] :
                                                                                  ( heap_T5999496708990702694on_nat
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( ( Succsum = none_nat )
                                                                                      = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                        = none_nat ) ) )
                                                                                  @ ^ [Va4: product_unit] :
                                                                                      ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                      @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ Info3 ) ) ) )
                  @ ( vEBT_c634343235235684882T_VEBT
                    @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                    @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                    @ T2 ) ) )
          @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( 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_5864_vebt__buildupi__refines,axiom,
    ! [N2: nat] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_V739175172307565963ildupi @ N2 ) ) ).

% vebt_buildupi_refines
thf(fact_5865_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T: vEBT_VEBT] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X2 ) ) ).

% vebt_inserti_refines
thf(fact_5866_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A4: nat,B4: nat] :
            ( P
            & ( Q @ A4 @ B4 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_5867_split__part,axiom,
    ! [P: $o,Q: int > int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [A4: int,B4: int] :
            ( P
            & ( Q @ A4 @ B4 ) ) )
      = ( ^ [Ab: product_prod_int_int] :
            ( P
            & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_5868_Collect__case__prod__mono,axiom,
    ! [A2: nat > nat > $o,B5: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ A2 @ B5 )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A2 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_5869_Collect__case__prod__mono,axiom,
    ! [A2: int > int > $o,B5: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A2 @ B5 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_5870_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu3: nat,Uv3: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_5871_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] :
      ( produc4947309494688390418_int_o
      @ ^ [Uu3: int,Uv3: int] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_5872_UNIV__def,axiom,
    ( top_top_set_int
    = ( collect_int
      @ ^ [X: int] : $true ) ) ).

% UNIV_def
thf(fact_5873_UNIV__def,axiom,
    ( top_top_set_complex
    = ( collect_complex
      @ ^ [X: complex] : $true ) ) ).

% UNIV_def
thf(fact_5874_UNIV__def,axiom,
    ( top_to4366644338036079209nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X: product_prod_int_int] : $true ) ) ).

% UNIV_def
thf(fact_5875_UNIV__def,axiom,
    ( top_top_set_real
    = ( collect_real
      @ ^ [X: real] : $true ) ) ).

% UNIV_def
thf(fact_5876_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_5877_UNIV__def,axiom,
    ( top_top_set_char
    = ( collect_char
      @ ^ [X: char] : $true ) ) ).

% UNIV_def
thf(fact_5878_VEBT_Ocase__distrib,axiom,
    ! [H2: produc819165548630102716T_VEBT > produc819165548630102716T_VEBT,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT,F22: $o > $o > produc819165548630102716T_VEBT,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_c634343235235684882T_VEBT @ F1 @ F22 @ VEBT ) )
      = ( vEBT_c634343235235684882T_VEBT
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_5879_VEBT_Ocase__distrib,axiom,
    ! [H2: produc819165548630102716T_VEBT > $o,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT,F22: $o > $o > produc819165548630102716T_VEBT,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_c634343235235684882T_VEBT @ F1 @ F22 @ VEBT ) )
      = ( vEBT_case_VEBT_o
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_5880_VEBT_Ocase__distrib,axiom,
    ! [H2: $o > produc819165548630102716T_VEBT,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o,F22: $o > $o > $o,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_case_VEBT_o @ F1 @ F22 @ VEBT ) )
      = ( vEBT_c634343235235684882T_VEBT
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_5881_VEBT_Ocase__distrib,axiom,
    ! [H2: $o > $o,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o,F22: $o > $o > $o,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_case_VEBT_o @ F1 @ F22 @ VEBT ) )
      = ( vEBT_case_VEBT_o
        @ ^ [X15: option4927543243414619207at_nat,X25: nat,X34: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_5882_is__Node__def,axiom,
    ( vEBT_is_Node
    = ( ^ [VEBT2: vEBT_VEBT] :
        ? [X113: option4927543243414619207at_nat,X123: nat,X133: list_VEBT_VEBT,X143: vEBT_VEBT] :
          ( VEBT2
          = ( vEBT_Node @ X113 @ X123 @ X133 @ X143 ) ) ) ) ).

% is_Node_def
thf(fact_5883_VEBT_OdiscI_I1_J,axiom,
    ! [VEBT: vEBT_VEBT,X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( VEBT
        = ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
     => ( vEBT_is_Node @ VEBT ) ) ).

% VEBT.discI(1)
thf(fact_5884_VEBT_Odisc_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : ( vEBT_is_Node @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBT.disc(1)
thf(fact_5885_VEBT_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT,F22: $o > $o > produc819165548630102716T_VEBT,X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_c634343235235684882T_VEBT @ F1 @ F22 @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBT.simps(5)
thf(fact_5886_VEBT_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o,F22: $o > $o > $o,X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_case_VEBT_o @ F1 @ F22 @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBT.simps(5)
thf(fact_5887_option_Othe__def,axiom,
    ( the_ui685118366354182287uint32
    = ( case_o6709414378691970003uint32 @ undefi332904144742839227uint32
      @ ^ [X25: ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32] : X25 ) ) ).

% option.the_def
thf(fact_5888_option_Othe__def,axiom,
    ( the_ui5136145761085816068eger_o
    = ( case_o4437601675458612413eger_o @ undefi6981832269580975664eger_o
      @ ^ [X25: ( uint32 > nat > $o ) > uint32 > code_integer > $o] : X25 ) ) ).

% option.the_def
thf(fact_5889_option_Othe__def,axiom,
    ( the_ui8720505876773817540uint32
    = ( case_o6228893485755354685uint32 @ undefi8537048349889504752uint32
      @ ^ [X25: ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32] : X25 ) ) ).

% option.the_def
thf(fact_5890_option_Othe__def,axiom,
    ( the_na2292640131888687716uint32
    = ( case_o8336680350232271869uint32 @ undefi7330133036835070352uint32
      @ ^ [X25: ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32] : X25 ) ) ).

% option.the_def
thf(fact_5891_option_Othe__def,axiom,
    ( the_na3915024202274359524uint32
    = ( case_o6516889040143735037uint32 @ undefi8952517107220742160uint32
      @ ^ [X25: ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32] : X25 ) ) ).

% option.the_def
thf(fact_5892_option_Othe__def,axiom,
    ( the_nat
    = ( case_option_nat_nat @ undefined_nat
      @ ^ [X25: nat] : X25 ) ) ).

% option.the_def
thf(fact_5893_option_Othe__def,axiom,
    ( the_Pr8591224930841456533at_nat
    = ( case_o7430979018509204427at_nat @ undefi3946296454836805481at_nat
      @ ^ [X25: product_prod_nat_nat] : X25 ) ) ).

% option.the_def
thf(fact_5894_option_Othe__def,axiom,
    ( the_num
    = ( case_option_num_num @ undefined_num
      @ ^ [X25: num] : X25 ) ) ).

% option.the_def
thf(fact_5895_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_5896_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_5897_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_5898_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_5899_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_5900_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_5901_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_5902_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,H4: heap_e7401611519738050253t_unit,R3: $o,N2: nat] :
      ( ! [Vebt_memberi2: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: $o,N8: nat] :
              ( ( heap_Time_effect_o @ ( Vebt_memberi2 @ A7 @ B7 @ Ba ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ Ba @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa3: $o,N4: nat] :
              ( ( heap_Time_effect_o
                @ ( vEBT_c6104975476656191286Heap_o
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T4 ) )
                      @ ^ [Uu3: 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 @ Deg4 @ 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
                                          @ ^ [Info4: option4927543243414619207at_nat] :
                                              ( produc5946672270950774087Heap_o
                                              @ ^ [Deg5: nat] :
                                                  ( produc5048428016959714504Heap_o
                                                  @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                                      ( heap_T7577940988442287570unit_o
                                                      @ ( refine_Imp_assert
                                                        @ ( ( Info3 = Info4 )
                                                          & ( Deg4 = Deg5 ) ) )
                                                      @ ^ [Uv3: product_unit] :
                                                          ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                          @ ^ [H: nat] :
                                                              ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                              @ ^ [L2: nat] :
                                                                  ( heap_T7577940988442287570unit_o
                                                                  @ ( refine_Imp_assert
                                                                    @ ( ( L2
                                                                        = ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                      & ( H
                                                                        = ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                                  @ ^ [Uw3: product_unit] :
                                                                      ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                                      @ ^ [Len: nat] :
                                                                          ( heap_T7577940988442287570unit_o
                                                                          @ ( refine_Imp_assert
                                                                            @ ( Len
                                                                              = ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                          @ ^ [Ux3: 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                                    & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                                                                                @ ^ [Uy3: product_unit] :
                                                                                    ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                                                    @ ^ [Th: vEBT_VEBTi] : ( Vebt_memberi2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                              @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                          @ ( vEBT_c634343235235684882T_VEBT
                                            @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                                            @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                                            @ T4 ) ) ) ) ) ) ) )
                          @ Info3 ) )
                  @ ^ [A4: $o,B4: $o] :
                      ( heap_Time_return_o
                      @ ( ( ( X3 = zero_zero_nat )
                         => A4 )
                        & ( ( X3 != zero_zero_nat )
                         => ( ( ( X3 = one_one_nat )
                             => B4 )
                            & ( X3 = one_one_nat ) ) ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T4 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_Time_effect_o @ ( produc5872130906356439992Heap_o @ ( produc2327743382103342416Heap_o @ vEBT_V854960066525838166emberi ) @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc1340562934675340024_nat_o @ ( produc3077134696498096400_nat_o @ P ) @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% VEBT_internal.vebt_memberi'.raw_induct
thf(fact_5903_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_5904_VEBT__internal_Ovebt__memberi_H_Osimps,axiom,
    ( vEBT_V854960066525838166emberi
    = ( ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6104975476656191286Heap_o
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
              @ ^ [Uu3: 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 @ Deg4 @ 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
                                  @ ^ [Info4: option4927543243414619207at_nat] :
                                      ( produc5946672270950774087Heap_o
                                      @ ^ [Deg5: nat] :
                                          ( produc5048428016959714504Heap_o
                                          @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                              ( heap_T7577940988442287570unit_o
                                              @ ( refine_Imp_assert
                                                @ ( ( Info3 = Info4 )
                                                  & ( Deg4 = Deg5 ) ) )
                                              @ ^ [Uv3: product_unit] :
                                                  ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [H: nat] :
                                                      ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [L2: nat] :
                                                          ( heap_T7577940988442287570unit_o
                                                          @ ( refine_Imp_assert
                                                            @ ( ( L2
                                                                = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                              & ( H
                                                                = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                          @ ^ [Uw3: product_unit] :
                                                              ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                              @ ^ [Len: nat] :
                                                                  ( heap_T7577940988442287570unit_o
                                                                  @ ( refine_Imp_assert
                                                                    @ ( Len
                                                                      = ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                  @ ^ [Ux3: 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                            & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                                                                        @ ^ [Uy3: product_unit] :
                                                                            ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                                            @ ^ [Th: vEBT_VEBTi] : ( vEBT_V854960066525838166emberi @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                      @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                  @ ( vEBT_c634343235235684882T_VEBT
                                    @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                                    @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                                    @ T2 ) ) ) ) ) ) ) )
                  @ Info3 ) )
          @ ^ [A4: $o,B4: $o] :
              ( heap_Time_return_o
              @ ( ( ( X = zero_zero_nat )
                 => A4 )
                & ( ( X != zero_zero_nat )
                 => ( ( ( X = one_one_nat )
                     => B4 )
                    & ( X = one_one_nat ) ) ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_memberi'.simps
thf(fact_5905_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,H4: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N2: nat] :
      ( ! [Vebt_inserti2: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: vEBT_VEBTi,N8: nat] :
              ( ( heap_T2071195472996403633_VEBTi @ ( Vebt_inserti2 @ A7 @ B7 @ Ba ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ Ba @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa3: vEBT_VEBTi,N4: nat] :
              ( ( heap_T2071195472996403633_VEBTi
                @ ( vEBT_c6028912655521741485_VEBTi
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                      @ ^ [Minma: product_prod_nat_nat] :
                          ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                          @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T4 ) )
                            @ ^ [Uu3: product_unit] :
                                ( produc2624314226134418078_VEBTi
                                @ ^ [Info4: option4927543243414619207at_nat] :
                                    ( produc7864503674192730076_VEBTi
                                    @ ^ [Deg5: nat] :
                                        ( produc9050507437146595227_VEBTi
                                        @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                            ( heap_T7982501707604696571_VEBTi
                                            @ ( refine_Imp_assert
                                              @ ( ( Info3 = Info4 )
                                                & ( Deg4 = Deg5 ) ) )
                                            @ ^ [Uv3: 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                    @ ^ [L2: nat] :
                                                                        ( heap_T7982501707604696571_VEBTi
                                                                        @ ( refine_Imp_assert
                                                                          @ ( L2
                                                                            = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X3 @ Mi4 ) @ Mi4 @ X3 ) @ ( divide_divide_nat @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                        @ ^ [Uw3: product_unit] :
                                                                            ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                            @ ^ [H: nat] :
                                                                                ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                                                @ ^ [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 @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                                      @ ^ [Ux3: product_unit] :
                                                                                          ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                                          @ ^ [Uy3: product_unit] :
                                                                                              ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                                                      @ ^ [Uz3: product_unit] :
                                                                                                          ( heap_T1006145433769338483_VEBTi @ ( Vebt_inserti2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                          @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                              ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                              @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                                  ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( Vebt_inserti2 @ Summary4 @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                                                                  @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                    @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                                @ ( the_Pr8591224930841456533at_nat @ Info4 ) ) ) ) )
                                @ ( vEBT_c634343235235684882T_VEBT
                                  @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                                  @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                                  @ T4 ) ) ) )
                      @ Info3 )
                  @ ^ [A4: $o,B4: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X3 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B4 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X3 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T4 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_T2071195472996403633_VEBTi @ ( produc2943724498215716011_VEBTi @ ( produc2298712477539903273_VEBTi @ vEBT_V3964819847710782039nserti ) @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc7403044070069621057_nat_o @ ( produc2677327216024927295_nat_o @ P ) @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% VEBT_internal.vebt_inserti'.raw_induct
thf(fact_5906_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,H4: heap_e7401611519738050253t_unit,R3: option_nat,N2: nat] :
      ( ! [Vebt_predi2: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: option_nat,N8: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_predi2 @ A7 @ B7 @ Ba ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ Ba @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa3: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T4 ) )
                      @ ^ [Uu3: product_unit] :
                          ( produc2190226783763740553on_nat
                          @ ^ [Info4: option4927543243414619207at_nat] :
                              ( produc2606485630176857543on_nat
                              @ ^ [Deg5: nat] :
                                  ( produc6124225815318652422on_nat
                                  @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                      ( heap_T5999496708990702694on_nat
                                      @ ( refine_Imp_assert
                                        @ ( ( Info4 = Info3 )
                                          & ( Deg5 = Deg4 )
                                          & ( vEBT_is_Node @ T4 ) ) )
                                      @ ^ [Uv3: product_unit] :
                                          ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                          @ ^ [Mima2: product_prod_nat_nat] :
                                              ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                                                @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [L2: nat] :
                                                      ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [H: nat] :
                                                          ( heap_T5999496708990702694on_nat
                                                          @ ( refine_Imp_assert
                                                            @ ( L2
                                                              = ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          @ ^ [Uw3: product_unit] :
                                                              ( heap_T5999496708990702694on_nat
                                                              @ ( refine_Imp_assert
                                                                @ ( H
                                                                  = ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              @ ^ [Ux3: product_unit] :
                                                                  ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                  @ ^ [Uy3: product_unit] :
                                                                      ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                              @ ^ [Uz3: product_unit] :
                                                                                  ( if_Hea5867803462524415986on_nat
                                                                                  @ ( ( Minlow != none_nat )
                                                                                    & ( vEBT_VEBT_greater @ ( some_nat @ L2 ) @ Minlow ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                    @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi2 @ Summary4 @ Summary3 @ H )
                                                                                    @ ^ [Predsum: option_nat] :
                                                                                        ( heap_T5999496708990702694on_nat
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( ( Predsum = none_nat )
                                                                                            = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                              = none_nat ) ) )
                                                                                        @ ^ [Va4: product_unit] :
                                                                                            ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                            @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                          @ Info3 ) ) ) )
                          @ ( vEBT_c634343235235684882T_VEBT
                            @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                            @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                            @ T4 ) ) )
                  @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X3 = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T4 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc183673358652719894on_nat @ ( produc1061038227461121684on_nat @ vEBT_VEBT_vebt_predi ) @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc8313044543888072982_nat_o @ ( produc275000359906850836_nat_o @ P ) @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% VEBT_internal.vebt_predi'.raw_induct
thf(fact_5907_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,H4: heap_e7401611519738050253t_unit,R3: option_nat,N2: nat] :
      ( ! [Vebt_succi2: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R6: option_nat,N8: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_succi2 @ A7 @ B7 @ Ba ) @ H6 @ H7 @ R6 @ N8 )
             => ( P @ A7 @ B7 @ Ba @ H6 @ H7 @ R6 @ N8 ) )
         => ! [T4: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa3: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T4 ) )
                      @ ^ [Uu3: product_unit] :
                          ( produc2190226783763740553on_nat
                          @ ^ [Info4: option4927543243414619207at_nat] :
                              ( produc2606485630176857543on_nat
                              @ ^ [Deg5: nat] :
                                  ( produc6124225815318652422on_nat
                                  @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                      ( heap_T5999496708990702694on_nat
                                      @ ( refine_Imp_assert
                                        @ ( ( Info4 = Info3 )
                                          & ( Deg5 = Deg4 )
                                          & ( vEBT_is_Node @ T4 ) ) )
                                      @ ^ [Uv3: product_unit] :
                                          ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                          @ ^ [Mima2: product_prod_nat_nat] :
                                              ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X3 @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                                                @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                  @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                    @ ^ [L2: nat] :
                                                        ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_T5999496708990702694on_nat
                                                            @ ( refine_Imp_assert
                                                              @ ( L2
                                                                = ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                            @ ^ [Uw3: product_unit] :
                                                                ( heap_T5999496708990702694on_nat
                                                                @ ( refine_Imp_assert
                                                                  @ ( H
                                                                    = ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                @ ^ [Ux3: product_unit] :
                                                                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                    @ ^ [Uy3: product_unit] :
                                                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                                @ ^ [Uz3: product_unit] :
                                                                                    ( if_Hea5867803462524415986on_nat
                                                                                    @ ( ( Maxlow != none_nat )
                                                                                      & ( vEBT_VEBT_less @ ( some_nat @ L2 ) @ Maxlow ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                      @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi2 @ Summary4 @ Summary3 @ H )
                                                                                      @ ^ [Succsum: option_nat] :
                                                                                          ( heap_T5999496708990702694on_nat
                                                                                          @ ( refine_Imp_assert
                                                                                            @ ( ( Succsum = none_nat )
                                                                                              = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                                = none_nat ) ) )
                                                                                          @ ^ [Va4: product_unit] :
                                                                                              ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                              @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                          @ Info3 ) ) ) )
                          @ ( vEBT_c634343235235684882T_VEBT
                            @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                            @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                            @ T4 ) ) )
                  @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( X3 = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T4 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc183673358652719894on_nat @ ( produc1061038227461121684on_nat @ vEBT_VEBT_vebt_succi ) @ Xa ) @ H2 @ H4 @ R3 @ N2 )
       => ( produc8313044543888072982_nat_o @ ( produc275000359906850836_nat_o @ P ) @ Xa @ H2 @ H4 @ R3 @ N2 ) ) ) ).

% VEBT_internal.vebt_succi'.raw_induct
thf(fact_5908_VEBT__internal_Ovebt__inserti_H_Osimps,axiom,
    ( vEBT_V3964819847710782039nserti
    = ( ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6028912655521741485_VEBTi
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
              @ ^ [Minma: product_prod_nat_nat] :
                  ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                  @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
                    @ ^ [Uu3: product_unit] :
                        ( produc2624314226134418078_VEBTi
                        @ ^ [Info4: option4927543243414619207at_nat] :
                            ( produc7864503674192730076_VEBTi
                            @ ^ [Deg5: nat] :
                                ( produc9050507437146595227_VEBTi
                                @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                    ( heap_T7982501707604696571_VEBTi
                                    @ ( refine_Imp_assert
                                      @ ( ( Info3 = Info4 )
                                        & ( Deg4 = Deg5 ) ) )
                                    @ ^ [Uv3: 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                            @ ^ [L2: nat] :
                                                                ( heap_T7982501707604696571_VEBTi
                                                                @ ( refine_Imp_assert
                                                                  @ ( L2
                                                                    = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide_nat @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                @ ^ [Uw3: product_unit] :
                                                                    ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                    @ ^ [H: nat] :
                                                                        ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                                        @ ^ [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 @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                              @ ^ [Ux3: product_unit] :
                                                                                  ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                                  @ ^ [Uy3: product_unit] :
                                                                                      ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                                              @ ^ [Uz3: product_unit] :
                                                                                                  ( heap_T1006145433769338483_VEBTi @ ( vEBT_V3964819847710782039nserti @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                  @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                      ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                      @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                          ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( vEBT_V3964819847710782039nserti @ Summary4 @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                                                          @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                            @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                        @ ( the_Pr8591224930841456533at_nat @ Info4 ) ) ) ) )
                        @ ( vEBT_c634343235235684882T_VEBT
                          @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                          @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                          @ T2 ) ) ) )
              @ Info3 )
          @ ^ [A4: $o,B4: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B4 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_inserti'.simps
thf(fact_5909_VEBT__internal_Ovebt__predi_H_Osimps,axiom,
    ( vEBT_VEBT_vebt_predi
    = ( ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
              @ ^ [Uu3: product_unit] :
                  ( produc2190226783763740553on_nat
                  @ ^ [Info4: option4927543243414619207at_nat] :
                      ( produc2606485630176857543on_nat
                      @ ^ [Deg5: nat] :
                          ( produc6124225815318652422on_nat
                          @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                              ( heap_T5999496708990702694on_nat
                              @ ( refine_Imp_assert
                                @ ( ( Info4 = Info3 )
                                  & ( Deg5 = Deg4 )
                                  & ( vEBT_is_Node @ T2 ) ) )
                              @ ^ [Uv3: product_unit] :
                                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                  @ ^ [Mima2: product_prod_nat_nat] :
                                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                                        @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                          @ ^ [L2: nat] :
                                              ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                              @ ^ [H: nat] :
                                                  ( heap_T5999496708990702694on_nat
                                                  @ ( refine_Imp_assert
                                                    @ ( L2
                                                      = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ^ [Uw3: product_unit] :
                                                      ( heap_T5999496708990702694on_nat
                                                      @ ( refine_Imp_assert
                                                        @ ( H
                                                          = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      @ ^ [Ux3: product_unit] :
                                                          ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                          @ ^ [Uy3: product_unit] :
                                                              ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                      @ ^ [Uz3: product_unit] :
                                                                          ( if_Hea5867803462524415986on_nat
                                                                          @ ( ( Minlow != none_nat )
                                                                            & ( vEBT_VEBT_greater @ ( some_nat @ L2 ) @ Minlow ) )
                                                                          @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_predi @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                            @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                          @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_predi @ Summary4 @ Summary3 @ H )
                                                                            @ ^ [Predsum: option_nat] :
                                                                                ( heap_T5999496708990702694on_nat
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( ( Predsum = none_nat )
                                                                                    = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                      = none_nat ) ) )
                                                                                @ ^ [Va4: product_unit] :
                                                                                    ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                    @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ Info3 ) ) ) )
                  @ ( vEBT_c634343235235684882T_VEBT
                    @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                    @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                    @ T2 ) ) )
          @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( 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_5910_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ( code_T8223577488154910793uint32
    = ( ^ [A_of_integer: code_integer > produc827990862158126777uint32,Zero: uint32,I2: code_integer] :
          ( if_uint32 @ ( ord_le6747313008572928689nteger @ I2 @ zero_z3403309356797280102nteger ) @ undefined_uint32
          @ ( if_uint32 @ ( I2 = zero_z3403309356797280102nteger ) @ Zero
            @ ( if_uint32
              @ ( ( modulo364778990260209775nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = zero_z3403309356797280102nteger )
              @ ( produc1510406741064981791uint32 @ ( A_of_integer @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) )
              @ ( produc9004433772639906525uint32 @ ( A_of_integer @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_5911_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ( code_T3174267988120873389le_nat
    = ( ^ [A_of_integer: code_integer > product_prod_nat_nat,Zero: nat,I2: code_integer] :
          ( if_nat @ ( ord_le6747313008572928689nteger @ I2 @ zero_z3403309356797280102nteger ) @ undefined_nat
          @ ( if_nat @ ( I2 = zero_z3403309356797280102nteger ) @ Zero
            @ ( if_nat
              @ ( ( modulo364778990260209775nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = zero_z3403309356797280102nteger )
              @ ( product_snd_nat_nat @ ( A_of_integer @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) )
              @ ( product_fst_nat_nat @ ( A_of_integer @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_5912_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ( code_T3171777517611823113le_int
    = ( ^ [A_of_integer: code_integer > product_prod_int_int,Zero: int,I2: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ I2 @ zero_z3403309356797280102nteger ) @ undefined_int
          @ ( if_int @ ( I2 = zero_z3403309356797280102nteger ) @ Zero
            @ ( if_int
              @ ( ( modulo364778990260209775nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = zero_z3403309356797280102nteger )
              @ ( product_snd_int_int @ ( A_of_integer @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) )
              @ ( product_fst_int_int @ ( A_of_integer @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_5913_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ( code_T5410181022262596154nteger
    = ( ^ [A_of_integer: code_integer > produc8923325533196201883nteger,Zero: code_integer,I2: code_integer] :
          ( if_Code_integer @ ( ord_le6747313008572928689nteger @ I2 @ zero_z3403309356797280102nteger ) @ undefi8781568570243851253nteger
          @ ( if_Code_integer @ ( I2 = zero_z3403309356797280102nteger ) @ Zero
            @ ( if_Code_integer
              @ ( ( modulo364778990260209775nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = zero_z3403309356797280102nteger )
              @ ( produc6174133586879617921nteger @ ( A_of_integer @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) )
              @ ( produc8508995932063986495nteger @ ( A_of_integer @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_5914_mod__h__bot__normalize,axiom,
    ! [H2: heap_e7401611519738050253t_unit,P: assn] :
      ( ( syntax5181832503971434150t_unit @ undefi332904144742839227uint32 @ H2 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ undefi8805113437130903861t_unit @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_normalize
thf(fact_5915_mod__h__bot__normalize,axiom,
    ! [H2: heap_e7401611519738050253t_unit,P: assn] :
      ( ( syntax3516980881688143089t_unit @ undefi6981832269580975664eger_o @ H2 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ undefi8805113437130903861t_unit @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_normalize
thf(fact_5916_mod__h__bot__normalize,axiom,
    ! [H2: heap_e7401611519738050253t_unit,P: assn] :
      ( ( syntax4084839521278773041t_unit @ undefi8537048349889504752uint32 @ H2 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ undefi8805113437130903861t_unit @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_normalize
thf(fact_5917_mod__h__bot__normalize,axiom,
    ! [H2: heap_e7401611519738050253t_unit,P: assn] :
      ( ( syntax2440797749228149777t_unit @ undefi7330133036835070352uint32 @ H2 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ undefi8805113437130903861t_unit @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_normalize
thf(fact_5918_mod__h__bot__normalize,axiom,
    ! [H2: heap_e7401611519738050253t_unit,P: assn] :
      ( ( syntax2089144231571168657t_unit @ undefi8952517107220742160uint32 @ H2 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ undefi8805113437130903861t_unit @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_normalize
thf(fact_5919_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_5920_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_5921_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_5922_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5923_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_5924_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_5925_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_5926_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_5927_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_5928_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_5929_numeral__div__numeral,axiom,
    ! [K: num,L: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ K @ L ) ) ) ).

% numeral_div_numeral
thf(fact_5930_numeral__div__numeral,axiom,
    ! [K: num,L: num] :
      ( ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ K ) @ ( numera6620942414471956472nteger @ L ) )
      = ( produc8508995932063986495nteger @ ( unique3479559517661332726nteger @ K @ L ) ) ) ).

% numeral_div_numeral
thf(fact_5931_numeral__div__numeral,axiom,
    ! [K: num,L: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) )
      = ( product_fst_int_int @ ( unique5052692396658037445od_int @ K @ L ) ) ) ).

% numeral_div_numeral
thf(fact_5932_numeral__mod__numeral,axiom,
    ! [K: num,L: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ K @ L ) ) ) ).

% numeral_mod_numeral
thf(fact_5933_numeral__mod__numeral,axiom,
    ! [K: num,L: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ K ) @ ( numera6620942414471956472nteger @ L ) )
      = ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ K @ L ) ) ) ).

% numeral_mod_numeral
thf(fact_5934_numeral__mod__numeral,axiom,
    ! [K: num,L: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) )
      = ( product_snd_int_int @ ( unique5052692396658037445od_int @ K @ L ) ) ) ).

% numeral_mod_numeral
thf(fact_5935_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_5936_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_5937_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_5938_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_5939_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_5940_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_5941_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_5942_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_5943_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_5944_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_5945_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_5946_one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( produc8508995932063986495nteger @ ( unique3479559517661332726nteger @ one @ N2 ) ) ) ).

% one_div_numeral
thf(fact_5947_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_5948_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_5949_one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ one @ N2 ) ) ) ).

% one_mod_numeral
thf(fact_5950_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_5951_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_5952_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_5953_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_5954_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_5955_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R3: int,Q6: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
       => ( R3 = R4 ) ) ) ).

% unique_remainder
thf(fact_5956_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R3: int,Q6: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
       => ( Q2 = Q6 ) ) ) ).

% unique_quotient
thf(fact_5957_VEBT_Odisc__eq__case_I1_J,axiom,
    ( vEBT_is_Node
    = ( vEBT_case_VEBT_o
      @ ^ [Uu3: option4927543243414619207at_nat,Uv3: nat,Uw3: list_VEBT_VEBT,Ux3: vEBT_VEBT] : $true
      @ ^ [Uu3: $o,Uv3: $o] : $false ) ) ).

% VEBT.disc_eq_case(1)
thf(fact_5958_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_5959_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R3 ) ) ).

% mod_int_unique
thf(fact_5960_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q2 ) ) ).

% div_int_unique
thf(fact_5961_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M7: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M7 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M7 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_5962_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_5963_fst__divmod,axiom,
    ! [M: num,N2: num] :
      ( ( produc8508995932063986495nteger @ ( unique3479559517661332726nteger @ M @ N2 ) )
      = ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% fst_divmod
thf(fact_5964_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_5965_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_5966_snd__divmod,axiom,
    ! [M: num,N2: num] :
      ( ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ M @ N2 ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% snd_divmod
thf(fact_5967_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_5968_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q2 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5969_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_5970_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M7: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M7 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M7 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_int_def
thf(fact_5971_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M7: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M7 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M7 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5972_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M7: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M7 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M7 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5973_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M7: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M7 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M7 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5974_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M7: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M7 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M7 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_5975_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R3 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
            & ( ord_less_int @ R3 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R3 )
                & ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5976_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M7: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M7 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M7 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M7 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5977_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M7: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M7 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M7 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M7 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5978_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M7: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M7 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M7 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M7 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5979_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5980_VEBT__internal_Ovebt__inserti_H_Omono,axiom,
    ! [X2: produc3960855945107176009Ti_nat] :
      ( comple5606513277678308283_VEBTi @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi ) @ heap_T7173139186834293313_VEBTi
      @ ^ [Vebt_inserti3: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
          ( produc2943724498215716011_VEBTi
          @ ( produc2298712477539903273_VEBTi
            @ ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6028912655521741485_VEBTi
                @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                    @ ^ [Minma: product_prod_nat_nat] :
                        ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
                          @ ^ [Uu3: product_unit] :
                              ( produc2624314226134418078_VEBTi
                              @ ^ [Info4: option4927543243414619207at_nat] :
                                  ( produc7864503674192730076_VEBTi
                                  @ ^ [Deg5: nat] :
                                      ( produc9050507437146595227_VEBTi
                                      @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                          ( heap_T7982501707604696571_VEBTi
                                          @ ( refine_Imp_assert
                                            @ ( ( Info3 = Info4 )
                                              & ( Deg4 = Deg5 ) ) )
                                          @ ^ [Uv3: 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                  @ ^ [L2: nat] :
                                                                      ( heap_T7982501707604696571_VEBTi
                                                                      @ ( refine_Imp_assert
                                                                        @ ( L2
                                                                          = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide_nat @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                      @ ^ [Uw3: product_unit] :
                                                                          ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                          @ ^ [H: nat] :
                                                                              ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                                              @ ^ [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 @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                                    @ ^ [Ux3: product_unit] :
                                                                                        ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                                        @ ^ [Uy3: product_unit] :
                                                                                            ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                                                    @ ^ [Uz3: product_unit] :
                                                                                                        ( heap_T1006145433769338483_VEBTi @ ( produc4062687810467144008_VEBTi @ ( produc2164094337957399884_VEBTi @ Vebt_inserti3 ) @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                        @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                            ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                            @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                                ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( produc4062687810467144008_VEBTi @ ( produc2164094337957399884_VEBTi @ Vebt_inserti3 ) @ Summary4 @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                                                                @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                  @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                              @ ( the_Pr8591224930841456533at_nat @ Info4 ) ) ) ) )
                              @ ( vEBT_c634343235235684882T_VEBT
                                @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                                @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                                @ T2 ) ) ) )
                    @ Info3 )
                @ ^ [A4: $o,B4: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B4 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_inserti'.mono
thf(fact_5981_divides__aux__eq,axiom,
    ! [Q2: nat,R3: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R3 ) )
      = ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_5982_divides__aux__eq,axiom,
    ! [Q2: int,R3: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R3 ) )
      = ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_5983_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_5984_flip__bit__Suc,axiom,
    ! [N2: nat,A: word_N3645301735248828278l_num1] :
      ( ( bit_se4491814353640558621l_num1 @ ( suc @ N2 ) @ A )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4491814353640558621l_num1 @ N2 @ ( divide1791077408188789448l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5985_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5986_flip__bit__Suc,axiom,
    ! [N2: nat,A: uint32] :
      ( ( bit_se7025624438249859091uint32 @ ( suc @ N2 ) @ A )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ N2 @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5987_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5988_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5989_unset__bit__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( bit_se5331074070815623765l_num1 @ zero_zero_nat @ A )
      = ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_5990_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_5991_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_5992_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
            @ ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6250501799366334488on_nat
                @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
                    @ ^ [Uu3: product_unit] :
                        ( produc2190226783763740553on_nat
                        @ ^ [Info4: option4927543243414619207at_nat] :
                            ( produc2606485630176857543on_nat
                            @ ^ [Deg5: nat] :
                                ( produc6124225815318652422on_nat
                                @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                    ( heap_T5999496708990702694on_nat
                                    @ ( refine_Imp_assert
                                      @ ( ( Info4 = Info3 )
                                        & ( Deg5 = Deg4 )
                                        & ( vEBT_is_Node @ T2 ) ) )
                                    @ ^ [Uv3: product_unit] :
                                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                        @ ^ [Mima2: product_prod_nat_nat] :
                                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [L2: nat] :
                                                      ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [H: nat] :
                                                          ( heap_T5999496708990702694on_nat
                                                          @ ( refine_Imp_assert
                                                            @ ( L2
                                                              = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          @ ^ [Uw3: product_unit] :
                                                              ( heap_T5999496708990702694on_nat
                                                              @ ( refine_Imp_assert
                                                                @ ( H
                                                                  = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              @ ^ [Ux3: product_unit] :
                                                                  ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                  @ ^ [Uy3: product_unit] :
                                                                      ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                              @ ^ [Uz3: product_unit] :
                                                                                  ( if_Hea5867803462524415986on_nat
                                                                                  @ ( ( Maxlow != none_nat )
                                                                                    & ( vEBT_VEBT_less @ ( some_nat @ L2 ) @ Maxlow ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_succi3 ) @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                    @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_succi3 ) @ Summary4 @ Summary3 @ H )
                                                                                    @ ^ [Succsum: option_nat] :
                                                                                        ( heap_T5999496708990702694on_nat
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( ( Succsum = none_nat )
                                                                                            = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                              = none_nat ) ) )
                                                                                        @ ^ [Va4: product_unit] :
                                                                                            ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                            @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                        @ Info3 ) ) ) )
                        @ ( vEBT_c634343235235684882T_VEBT
                          @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                          @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                          @ T2 ) ) )
                @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( 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_5993_dvd__0__right,axiom,
    ! [A: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A @ zero_z3563351764282998399l_num1 ) ).

% dvd_0_right
thf(fact_5994_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_5995_dvd__0__right,axiom,
    ! [A: uint32] : ( dvd_dvd_uint32 @ A @ zero_zero_uint32 ) ).

% dvd_0_right
thf(fact_5996_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_5997_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_5998_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_5999_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_6000_dvd__0__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ zero_z3563351764282998399l_num1 @ A )
      = ( A = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left_iff
thf(fact_6001_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_6002_dvd__0__left__iff,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A )
      = ( A = zero_zero_uint32 ) ) ).

% dvd_0_left_iff
thf(fact_6003_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_6004_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_6005_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_6006_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_6007_dvd__add__triv__right__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ A @ ( plus_plus_uint32 @ B @ A ) )
      = ( dvd_dvd_uint32 @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6008_dvd__add__triv__right__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ ( plus_p361126936061061375l_num1 @ B @ A ) )
      = ( dvd_dv6812691276156420380l_num1 @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6009_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6010_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6011_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6012_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6013_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6014_dvd__add__triv__left__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ A @ ( plus_plus_uint32 @ A @ B ) )
      = ( dvd_dvd_uint32 @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6015_dvd__add__triv__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ ( plus_p361126936061061375l_num1 @ A @ B ) )
      = ( dvd_dv6812691276156420380l_num1 @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6016_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6017_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6018_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6019_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6020_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6021_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_6022_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_6023_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_6024_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_6025_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_6026_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_6027_curry__conv,axiom,
    ( produc1114182431767986483on_nat
    = ( ^ [F4: produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat,A4: vEBT_VEBT,B4: vEBT_VEBTi] : ( F4 @ ( produc6084888613844515218_VEBTi @ A4 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_6028_curry__conv,axiom,
    ( produc2663629013181010545Heap_o
    = ( ^ [F4: produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o,A4: vEBT_VEBT,B4: vEBT_VEBTi] : ( F4 @ ( produc6084888613844515218_VEBTi @ A4 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_6029_curry__conv,axiom,
    ( produc2164094337957399884_VEBTi
    = ( ^ [F4: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi,A4: produc3625547720036274456_VEBTi,B4: nat] : ( F4 @ ( produc1853644041309157249Ti_nat @ A4 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_6030_curry__conv,axiom,
    ( produc1757988346207259447on_nat
    = ( ^ [F4: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat,A4: produc3625547720036274456_VEBTi,B4: nat] : ( F4 @ ( produc1853644041309157249Ti_nat @ A4 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_6031_curry__conv,axiom,
    ( produc8381543706267210711Heap_o
    = ( ^ [F4: produc3960855945107176009Ti_nat > heap_Time_Heap_o,A4: produc3625547720036274456_VEBTi,B4: nat] : ( F4 @ ( produc1853644041309157249Ti_nat @ A4 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_6032_case__prod__curry,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat] :
      ( ( produc1061038227461121684on_nat @ ( produc1114182431767986483on_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6033_case__prod__curry,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o] :
      ( ( produc2327743382103342416Heap_o @ ( produc2663629013181010545Heap_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6034_case__prod__curry,axiom,
    ! [F: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
      ( ( produc2943724498215716011_VEBTi @ ( produc2164094337957399884_VEBTi @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6035_case__prod__curry,axiom,
    ! [F: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
      ( ( produc183673358652719894on_nat @ ( produc1757988346207259447on_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6036_case__prod__curry,axiom,
    ! [F: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
      ( ( produc5872130906356439992Heap_o @ ( produc8381543706267210711Heap_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6037_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ ( produc6629854527392350932at_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6038_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o @ ( produc1310100445399344235_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6039_case__prod__curry,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ ( produc8249235968001453780nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6040_case__prod__curry,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o @ ( produc175634133007206835_int_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6041_case__prod__curry,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int @ ( produc1016772743285680337nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6042_curry__case__prod,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
      ( ( produc1114182431767986483on_nat @ ( produc1061038227461121684on_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6043_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_6044_curry__case__prod,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
      ( ( produc2164094337957399884_VEBTi @ ( produc2943724498215716011_VEBTi @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6045_curry__case__prod,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat] :
      ( ( produc1757988346207259447on_nat @ ( produc183673358652719894on_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6046_curry__case__prod,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o] :
      ( ( produc8381543706267210711Heap_o @ ( produc5872130906356439992Heap_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6047_curry__case__prod,axiom,
    ! [F: nat > nat > product_prod_nat_nat] :
      ( ( produc6629854527392350932at_nat @ ( produc2626176000494625587at_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6048_curry__case__prod,axiom,
    ! [F: nat > nat > $o] :
      ( ( produc1310100445399344235_nat_o @ ( produc6081775807080527818_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6049_curry__case__prod,axiom,
    ! [F: int > int > product_prod_int_int] :
      ( ( produc8249235968001453780nt_int @ ( produc4245557441103728435nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6050_curry__case__prod,axiom,
    ! [F: int > int > $o] :
      ( ( produc175634133007206835_int_o @ ( produc4947309494688390418_int_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6051_curry__case__prod,axiom,
    ! [F: int > int > int] :
      ( ( produc1016772743285680337nt_int @ ( produc8211389475949308722nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6052_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_6053_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_6054_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_6055_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_6056_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_6057_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_6058_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_6059_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_6060_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_6061_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_6062_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_6063_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_6064_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_6065_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_6066_dvd__add__times__triv__left__iff,axiom,
    ! [A: uint32,C: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ A @ ( plus_plus_uint32 @ ( times_times_uint32 @ C @ A ) @ B ) )
      = ( dvd_dvd_uint32 @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6067_dvd__add__times__triv__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ C @ A ) @ B ) )
      = ( dvd_dv6812691276156420380l_num1 @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6068_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6069_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6070_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6071_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6072_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6073_dvd__add__times__triv__right__iff,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A @ ( plus_plus_uint32 @ B @ ( times_times_uint32 @ C @ A ) ) )
      = ( dvd_dvd_uint32 @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6074_dvd__add__times__triv__right__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ ( plus_p361126936061061375l_num1 @ B @ ( times_7065122842183080059l_num1 @ C @ A ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6075_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6076_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6077_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6078_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6079_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6080_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_6081_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_6082_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_6083_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_6084_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_6085_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_6086_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_6087_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_6088_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_6089_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_6090_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_6091_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_6092_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_6093_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_6094_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_6095_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_6096_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_6097_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_6098_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_6099_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_6100_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_6101_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_6102_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_6103_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_6104_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_6105_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_6106_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_6107_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_6108_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_6109_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_6110_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_6111_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_6112_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_6113_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_6114_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_6115_odd__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6116_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6117_odd__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A @ B ) ) )
      = ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6118_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6119_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6120_even__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6121_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6122_even__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A @ B ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
        = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6123_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6124_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6125_even__mult__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( times_times_uint32 @ A @ B ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_6126_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_6127_even__mult__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( times_7065122842183080059l_num1 @ A @ B ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
        | ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_6128_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_6129_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_6130_even__mod__2__iff,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_6131_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_6132_even__mod__2__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_6133_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_6134_even__mod__2__iff,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_6135_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_6136_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_6137_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_6138_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_6139_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_6140_even__plus__one__iff,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ one_one_uint32 ) )
      = ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6141_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6142_even__plus__one__iff,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A @ one_on7727431528512463931l_num1 ) )
      = ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6143_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6144_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6145_even__diff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_uint32 @ A @ B ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) ) ) ).

% even_diff
thf(fact_6146_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% even_diff
thf(fact_6147_even__diff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_4019991460397169231l_num1 @ A @ B ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A @ B ) ) ) ).

% even_diff
thf(fact_6148_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_6149_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_6150_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_6151_even__succ__div__2,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A ) @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6152_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6153_even__succ__div__2,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
     => ( ( divide1791077408188789448l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = ( divide1791077408188789448l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6154_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6155_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6156_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_6157_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_6158_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_6159_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6160_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6161_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6162_even__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_6163_even__power,axiom,
    ! [A: uint32,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( power_power_uint32 @ A @ N2 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_6164_even__power,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( power_2184487114949457152l_num1 @ A @ N2 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_6165_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_6166_even__power,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_6167_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_6168_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_6169_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_6170_power__less__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_6171_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_6172_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_6173_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_6174_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_6175_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_6176_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_6177_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_6178_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_6179_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_6180_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_6181_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_6182_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_6183_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_6184_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_6185_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_6186_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_6187_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_6188_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_6189_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_6190_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_6191_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_6192_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_6193_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_6194_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_6195_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_6196_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_6197_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_6198_even__succ__div__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6199_even__succ__div__exp,axiom,
    ! [A: uint32,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6200_even__succ__div__exp,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide1791077408188789448l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide1791077408188789448l_num1 @ A @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6201_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6202_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6203_even__succ__mod__exp,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo1504961113040953224l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( modulo1504961113040953224l_num1 @ A @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6204_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6205_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6206_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6207_even__succ__mod__exp,axiom,
    ! [A: uint32,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_uint32 @ one_one_uint32 @ ( modulo_modulo_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6208_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_6209_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_6210_dvd__refl,axiom,
    ! [A: uint32] : ( dvd_dvd_uint32 @ A @ A ) ).

% dvd_refl
thf(fact_6211_dvd__refl,axiom,
    ! [A: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A @ A ) ).

% dvd_refl
thf(fact_6212_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_6213_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_6214_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_6215_dvd__trans,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A @ B )
     => ( ( dvd_dvd_uint32 @ B @ C )
       => ( dvd_dvd_uint32 @ A @ C ) ) ) ).

% dvd_trans
thf(fact_6216_dvd__trans,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ B )
     => ( ( dvd_dv6812691276156420380l_num1 @ B @ C )
       => ( dvd_dv6812691276156420380l_num1 @ A @ C ) ) ) ).

% dvd_trans
thf(fact_6217_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_6218_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_6219_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_6220_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_6221_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_6222_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_6223_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_6224_curry__K,axiom,
    ! [C: heap_T8145700208782473153_VEBTi] :
      ( ( produc2164094337957399884_VEBTi
        @ ^ [X: produc3960855945107176009Ti_nat] : C )
      = ( ^ [X: produc3625547720036274456_VEBTi,Y: nat] : C ) ) ).

% curry_K
thf(fact_6225_curry__K,axiom,
    ! [C: heap_T2636463487746394924on_nat] :
      ( ( produc1757988346207259447on_nat
        @ ^ [X: produc3960855945107176009Ti_nat] : C )
      = ( ^ [X: produc3625547720036274456_VEBTi,Y: nat] : C ) ) ).

% curry_K
thf(fact_6226_curry__K,axiom,
    ! [C: heap_Time_Heap_o] :
      ( ( produc8381543706267210711Heap_o
        @ ^ [X: produc3960855945107176009Ti_nat] : C )
      = ( ^ [X: produc3625547720036274456_VEBTi,Y: nat] : C ) ) ).

% curry_K
thf(fact_6227_dvd__0__left,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ zero_z3563351764282998399l_num1 @ A )
     => ( A = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left
thf(fact_6228_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_6229_dvd__0__left,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A )
     => ( A = zero_zero_uint32 ) ) ).

% dvd_0_left
thf(fact_6230_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_6231_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_6232_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_6233_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_6234_dvd__add__right__iff,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A @ B )
     => ( ( dvd_dvd_uint32 @ A @ ( plus_plus_uint32 @ B @ C ) )
        = ( dvd_dvd_uint32 @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6235_dvd__add__right__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ B )
     => ( ( dvd_dv6812691276156420380l_num1 @ A @ ( plus_p361126936061061375l_num1 @ B @ C ) )
        = ( dvd_dv6812691276156420380l_num1 @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6236_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6237_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6238_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6239_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6240_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6241_dvd__add__left__iff,axiom,
    ! [A: uint32,C: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ A @ C )
     => ( ( dvd_dvd_uint32 @ A @ ( plus_plus_uint32 @ B @ C ) )
        = ( dvd_dvd_uint32 @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6242_dvd__add__left__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ C )
     => ( ( dvd_dv6812691276156420380l_num1 @ A @ ( plus_p361126936061061375l_num1 @ B @ C ) )
        = ( dvd_dv6812691276156420380l_num1 @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6243_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6244_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6245_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6246_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6247_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6248_dvd__add,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A @ B )
     => ( ( dvd_dvd_uint32 @ A @ C )
       => ( dvd_dvd_uint32 @ A @ ( plus_plus_uint32 @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6249_dvd__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ B )
     => ( ( dvd_dv6812691276156420380l_num1 @ A @ C )
       => ( dvd_dv6812691276156420380l_num1 @ A @ ( plus_p361126936061061375l_num1 @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6250_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6251_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6252_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6253_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6254_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6255_one__dvd,axiom,
    ! [A: uint32] : ( dvd_dvd_uint32 @ one_one_uint32 @ A ) ).

% one_dvd
thf(fact_6256_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_6257_one__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ one_on7727431528512463931l_num1 @ A ) ).

% one_dvd
thf(fact_6258_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_6259_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_6260_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_6261_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_6262_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_6263_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_6264_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_6265_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_6266_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_6267_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_6268_dvdE,axiom,
    ! [B: uint32,A: uint32] :
      ( ( dvd_dvd_uint32 @ B @ A )
     => ~ ! [K3: uint32] :
            ( A
           != ( times_times_uint32 @ B @ K3 ) ) ) ).

% dvdE
thf(fact_6269_dvdE,axiom,
    ! [B: word_N3645301735248828278l_num1,A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ B @ A )
     => ~ ! [K3: word_N3645301735248828278l_num1] :
            ( A
           != ( times_7065122842183080059l_num1 @ B @ K3 ) ) ) ).

% dvdE
thf(fact_6270_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K3: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).

% dvdE
thf(fact_6271_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K3: real] :
            ( A
           != ( times_times_real @ B @ K3 ) ) ) ).

% dvdE
thf(fact_6272_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K3: rat] :
            ( A
           != ( times_times_rat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_6273_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K3: nat] :
            ( A
           != ( times_times_nat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_6274_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K3: int] :
            ( A
           != ( times_times_int @ B @ K3 ) ) ) ).

% dvdE
thf(fact_6275_dvdI,axiom,
    ! [A: uint32,B: uint32,K: uint32] :
      ( ( A
        = ( times_times_uint32 @ B @ K ) )
     => ( dvd_dvd_uint32 @ B @ A ) ) ).

% dvdI
thf(fact_6276_dvdI,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( A
        = ( times_7065122842183080059l_num1 @ B @ K ) )
     => ( dvd_dv6812691276156420380l_num1 @ B @ A ) ) ).

% dvdI
thf(fact_6277_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B @ K ) )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% dvdI
thf(fact_6278_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_6279_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_6280_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_6281_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_6282_dvd__def,axiom,
    ( dvd_dvd_uint32
    = ( ^ [B4: uint32,A4: uint32] :
        ? [K4: uint32] :
          ( A4
          = ( times_times_uint32 @ B4 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_6283_dvd__def,axiom,
    ( dvd_dv6812691276156420380l_num1
    = ( ^ [B4: word_N3645301735248828278l_num1,A4: word_N3645301735248828278l_num1] :
        ? [K4: word_N3645301735248828278l_num1] :
          ( A4
          = ( times_7065122842183080059l_num1 @ B4 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_6284_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B4: code_integer,A4: code_integer] :
        ? [K4: code_integer] :
          ( A4
          = ( times_3573771949741848930nteger @ B4 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_6285_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B4: real,A4: real] :
        ? [K4: real] :
          ( A4
          = ( times_times_real @ B4 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_6286_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B4: rat,A4: rat] :
        ? [K4: rat] :
          ( A4
          = ( times_times_rat @ B4 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_6287_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B4: nat,A4: nat] :
        ? [K4: nat] :
          ( A4
          = ( times_times_nat @ B4 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_6288_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B4: int,A4: int] :
        ? [K4: int] :
          ( A4
          = ( times_times_int @ B4 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_6289_dvd__mult,axiom,
    ! [A: uint32,C: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ A @ C )
     => ( dvd_dvd_uint32 @ A @ ( times_times_uint32 @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6290_dvd__mult,axiom,
    ! [A: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ C )
     => ( dvd_dv6812691276156420380l_num1 @ A @ ( times_7065122842183080059l_num1 @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6291_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6292_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6293_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6294_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6295_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6296_dvd__mult2,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A @ B )
     => ( dvd_dvd_uint32 @ A @ ( times_times_uint32 @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6297_dvd__mult2,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ B )
     => ( dvd_dv6812691276156420380l_num1 @ A @ ( times_7065122842183080059l_num1 @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6298_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6299_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6300_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6301_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6302_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6303_dvd__mult__left,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ ( times_times_uint32 @ A @ B ) @ C )
     => ( dvd_dvd_uint32 @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6304_dvd__mult__left,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A @ B ) @ C )
     => ( dvd_dv6812691276156420380l_num1 @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6305_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6306_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6307_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6308_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6309_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6310_dvd__triv__left,axiom,
    ! [A: uint32,B: uint32] : ( dvd_dvd_uint32 @ A @ ( times_times_uint32 @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6311_dvd__triv__left,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A @ ( times_7065122842183080059l_num1 @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6312_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6313_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6314_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6315_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6316_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6317_mult__dvd__mono,axiom,
    ! [A: uint32,B: uint32,C: uint32,D: uint32] :
      ( ( dvd_dvd_uint32 @ A @ B )
     => ( ( dvd_dvd_uint32 @ C @ D )
       => ( dvd_dvd_uint32 @ ( times_times_uint32 @ A @ C ) @ ( times_times_uint32 @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6318_mult__dvd__mono,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,D: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A @ B )
     => ( ( dvd_dv6812691276156420380l_num1 @ C @ D )
       => ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A @ C ) @ ( times_7065122842183080059l_num1 @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6319_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6320_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6321_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6322_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6323_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6324_dvd__mult__right,axiom,
    ! [A: uint32,B: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ ( times_times_uint32 @ A @ B ) @ C )
     => ( dvd_dvd_uint32 @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6325_dvd__mult__right,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A @ B ) @ C )
     => ( dvd_dv6812691276156420380l_num1 @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6326_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6327_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6328_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6329_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6330_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6331_dvd__triv__right,axiom,
    ! [A: uint32,B: uint32] : ( dvd_dvd_uint32 @ A @ ( times_times_uint32 @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6332_dvd__triv__right,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A @ ( times_7065122842183080059l_num1 @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6333_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6334_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6335_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6336_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6337_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6338_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_6339_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_6340_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_6341_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_6342_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_6343_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_6344_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_6345_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_6346_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( ( divide6298287555418463151nteger @ A @ C )
            = ( divide6298287555418463151nteger @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6347_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6348_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6349_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6350_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6351_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ C )
        = ( divide6298287555418463151nteger @ B @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A )
       => ( ( dvd_dvd_Code_integer @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6352_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6353_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6354_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6355_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6356_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_6357_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_6358_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_6359_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_6360_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_6361_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_6362_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_6363_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_6364_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_6365_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_6366_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_6367_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_6368_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6369_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6370_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6371_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_6372_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_6373_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_6374_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_6375_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_6376_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_6377_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_6378_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_6379_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6380_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6381_subset__divisors__dvd,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ord_le2219237028632753026uint32
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ A ) )
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ B ) ) )
      = ( dvd_dvd_uint32 @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6382_subset__divisors__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( ord_le5203802739334966412l_num1
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ A ) )
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ B ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6383_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6384_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6385_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6386_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6387_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6388_strict__subset__divisors__dvd,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ord_less_set_uint32
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ A ) )
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ B ) ) )
      = ( ( dvd_dvd_uint32 @ A @ B )
        & ~ ( dvd_dvd_uint32 @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6389_strict__subset__divisors__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( ord_le6726900395242856064l_num1
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ A ) )
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ B ) ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ A @ B )
        & ~ ( dvd_dv6812691276156420380l_num1 @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6390_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B )
        & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6391_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_6392_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_6393_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_6394_minf_I10_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_6395_minf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X6 @ Z3 )
     => ( ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_6396_minf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_6397_minf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_6398_minf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_6399_minf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_6400_minf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_6401_minf_I9_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ X6 @ Z3 )
     => ( ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) )
        = ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_6402_minf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X6 @ Z3 )
     => ( ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) )
        = ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_6403_minf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X6 @ Z3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_6404_minf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_6405_minf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_6406_minf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_6407_minf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_6408_pinf_I10_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_6409_pinf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Z3 @ X6 )
     => ( ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_6410_pinf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_6411_pinf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_6412_pinf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_6413_pinf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_6414_pinf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_6415_pinf_I9_J,axiom,
    ! [D: uint32,S3: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ Z3 @ X6 )
     => ( ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) )
        = ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_6416_pinf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S3: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Z3 @ X6 )
     => ( ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) )
        = ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_6417_pinf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X6 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_6418_pinf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_6419_pinf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_6420_pinf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_6421_pinf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_6422_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6423_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( ( divide_divide_real @ A @ B )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6424_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( ( divide_divide_rat @ A @ B )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6425_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6426_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6427_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_6428_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_6429_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_6430_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6431_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6432_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6433_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6434_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6435_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6436_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6437_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6438_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6439_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6440_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6441_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6442_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6443_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6444_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6445_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6446_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6447_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6448_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6449_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6450_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6451_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6452_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6453_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6454_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_6455_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_6456_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_6457_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_6458_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_6459_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_6460_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_6461_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_6462_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_6463_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6464_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6465_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6466_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6467_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6468_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6469_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6470_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6471_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6472_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6473_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6474_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6475_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6476_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6477_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6478_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6479_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6480_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6481_div__power,axiom,
    ! [B: code_integer,A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N2 )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_6482_div__power,axiom,
    ! [B: nat,A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N2 )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_6483_div__power,axiom,
    ! [B: int,A: int,N2: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N2 )
        = ( divide_divide_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_6484_mod__0__imp__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A @ B )
        = zero_z3563351764282998399l_num1 )
     => ( dvd_dv6812691276156420380l_num1 @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_6485_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_6486_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_6487_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_6488_mod__0__imp__dvd,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( modulo_modulo_uint32 @ A @ B )
        = zero_zero_uint32 )
     => ( dvd_dvd_uint32 @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_6489_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( modulo_modulo_nat @ B4 @ A4 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6490_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A4: int,B4: int] :
          ( ( modulo_modulo_int @ B4 @ A4 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6491_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A4: code_integer,B4: code_integer] :
          ( ( modulo364778990260209775nteger @ B4 @ A4 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6492_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_6493_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_6494_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_6495_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_6496_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_6497_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_6498_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_6499_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_6500_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_6501_power__le__dvd,axiom,
    ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6502_power__le__dvd,axiom,
    ! [A: uint32,N2: nat,B: uint32,M: nat] :
      ( ( dvd_dvd_uint32 @ ( power_power_uint32 @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_uint32 @ ( power_power_uint32 @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6503_power__le__dvd,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat,B: word_N3645301735248828278l_num1,M: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6504_power__le__dvd,axiom,
    ! [A: real,N2: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6505_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6506_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6507_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6508_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: uint32] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_uint32 @ ( power_power_uint32 @ A @ M ) @ ( power_power_uint32 @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6509_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A @ M ) @ ( power_2184487114949457152l_num1 @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6510_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6511_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6512_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6513_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_6514_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_6515_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_6516_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_6517_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_6518_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_6519_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_6520_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_6521_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_6522_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_6523_even__unset__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ M @ A ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6524_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6525_even__unset__bit__iff,axiom,
    ! [M: nat,A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se5331074070815623765l_num1 @ M @ A ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6526_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6527_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6528_even__flip__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ M @ A ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6529_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6530_even__flip__bit__iff,axiom,
    ! [M: nat,A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4491814353640558621l_num1 @ M @ A ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6531_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6532_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6533_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_6534_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_6535_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_6536_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_6537_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_6538_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_6539_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_6540_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_6541_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_6542_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_6543_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_6544_even__numeral,axiom,
    ! [N2: num] : ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_6545_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_6546_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_6547_unity__coeff__ex,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,L: word_N3645301735248828278l_num1] :
      ( ( ? [X: word_N3645301735248828278l_num1] : ( P @ ( times_7065122842183080059l_num1 @ L @ X ) ) )
      = ( ? [X: word_N3645301735248828278l_num1] :
            ( ( dvd_dv6812691276156420380l_num1 @ L @ ( plus_p361126936061061375l_num1 @ X @ zero_z3563351764282998399l_num1 ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_6548_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
      = ( ? [X: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_6549_unity__coeff__ex,axiom,
    ! [P: uint32 > $o,L: uint32] :
      ( ( ? [X: uint32] : ( P @ ( times_times_uint32 @ L @ X ) ) )
      = ( ? [X: uint32] :
            ( ( dvd_dvd_uint32 @ L @ ( plus_plus_uint32 @ X @ zero_zero_uint32 ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_6550_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
      = ( ? [X: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_6551_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
      = ( ? [X: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_6552_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
      = ( ? [X: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_6553_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
      = ( ? [X: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_6554_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_6555_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_6556_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_6557_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_6558_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_6559_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_6560_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B )
       => ( ( ( divide6298287555418463151nteger @ B @ A )
            = C )
          = ( B
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6561_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6562_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6563_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6564_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6565_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6566_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6567_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6568_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6569_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6570_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6571_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6572_inf__period_I4_J,axiom,
    ! [D: uint32,D5: uint32,T: uint32] :
      ( ( dvd_dvd_uint32 @ D @ D5 )
     => ! [X6: uint32,K5: uint32] :
          ( ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ ( minus_minus_uint32 @ X6 @ ( times_times_uint32 @ K5 @ D5 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6573_inf__period_I4_J,axiom,
    ! [D: word_N3645301735248828278l_num1,D5: word_N3645301735248828278l_num1,T: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ D @ D5 )
     => ! [X6: word_N3645301735248828278l_num1,K5: word_N3645301735248828278l_num1] :
          ( ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ T ) ) )
          = ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ ( minus_4019991460397169231l_num1 @ X6 @ ( times_7065122842183080059l_num1 @ K5 @ D5 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6574_inf__period_I4_J,axiom,
    ! [D: code_integer,D5: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D5 )
     => ! [X6: code_integer,K5: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K5 @ D5 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6575_inf__period_I4_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( ( dvd_dvd_real @ D @ D5 )
     => ! [X6: real,K5: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K5 @ D5 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6576_inf__period_I4_J,axiom,
    ! [D: rat,D5: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D5 )
     => ! [X6: rat,K5: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K5 @ D5 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6577_inf__period_I4_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( ( dvd_dvd_int @ D @ D5 )
     => ! [X6: int,K5: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K5 @ D5 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6578_inf__period_I3_J,axiom,
    ! [D: uint32,D5: uint32,T: uint32] :
      ( ( dvd_dvd_uint32 @ D @ D5 )
     => ! [X6: uint32,K5: uint32] :
          ( ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ T ) )
          = ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ ( minus_minus_uint32 @ X6 @ ( times_times_uint32 @ K5 @ D5 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6579_inf__period_I3_J,axiom,
    ! [D: word_N3645301735248828278l_num1,D5: word_N3645301735248828278l_num1,T: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ D @ D5 )
     => ! [X6: word_N3645301735248828278l_num1,K5: word_N3645301735248828278l_num1] :
          ( ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ T ) )
          = ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ ( minus_4019991460397169231l_num1 @ X6 @ ( times_7065122842183080059l_num1 @ K5 @ D5 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6580_inf__period_I3_J,axiom,
    ! [D: code_integer,D5: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D5 )
     => ! [X6: code_integer,K5: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K5 @ D5 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6581_inf__period_I3_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( ( dvd_dvd_real @ D @ D5 )
     => ! [X6: real,K5: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K5 @ D5 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6582_inf__period_I3_J,axiom,
    ! [D: rat,D5: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D5 )
     => ! [X6: rat,K5: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K5 @ D5 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6583_inf__period_I3_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( ( dvd_dvd_int @ D @ D5 )
     => ! [X6: int,K5: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K5 @ D5 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6584_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_6585_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_6586_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_6587_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6588_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6589_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6590_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6591_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6592_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6593_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_6594_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_6595_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_6596_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6597_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6598_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6599_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6600_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6601_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6602_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_6603_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_6604_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_6605_is__unit__power__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6606_is__unit__power__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6607_is__unit__power__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6608_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_6609_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_6610_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_6611_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_6612_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_6613_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_6614_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_6615_even__zero,axiom,
    dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ zero_zero_uint32 ).

% even_zero
thf(fact_6616_even__zero,axiom,
    dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ zero_z3563351764282998399l_num1 ).

% even_zero
thf(fact_6617_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_6618_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_6619_odd__even__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6620_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6621_odd__even__add,axiom,
    ! [A: word_N3645301735248828278l_num1,B: word_N3645301735248828278l_num1] :
      ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B )
       => ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6622_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6623_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6624_odd__one,axiom,
    ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ one_one_uint32 ) ).

% odd_one
thf(fact_6625_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_6626_odd__one,axiom,
    ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ one_on7727431528512463931l_num1 ) ).

% odd_one
thf(fact_6627_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_6628_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_6629_evenE,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: uint32] :
            ( A
           != ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_6630_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_6631_evenE,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: word_N3645301735248828278l_num1] :
            ( A
           != ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_6632_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_6633_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_6634_bit__eq__rec,axiom,
    ( ( ^ [Y5: uint32,Z4: uint32] : Y5 = Z4 )
    = ( ^ [A4: uint32,B4: uint32] :
          ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_uint32 @ A4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
            = ( divide_divide_uint32 @ B4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6635_bit__eq__rec,axiom,
    ( ( ^ [Y5: code_integer,Z4: code_integer] : Y5 = Z4 )
    = ( ^ [A4: code_integer,B4: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6636_bit__eq__rec,axiom,
    ( ( ^ [Y5: word_N3645301735248828278l_num1,Z4: word_N3645301735248828278l_num1] : Y5 = Z4 )
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] :
          ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide1791077408188789448l_num1 @ A4 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
            = ( divide1791077408188789448l_num1 @ B4 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6637_bit__eq__rec,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6638_bit__eq__rec,axiom,
    ( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
    = ( ^ [A4: int,B4: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6639_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B2 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B2 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6640_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B2 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B2 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B2 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6641_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( ( dvd_dvd_int @ B2 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B2 )
                 => ( ( ( divide_divide_int @ one_one_int @ B2 )
                      = A )
                   => ( ( ( times_times_int @ A @ B2 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6642_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_6643_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_6644_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_6645_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_6646_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_6647_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_6648_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6649_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6650_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( numera7442385471795722001l_num1 @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6651_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6652_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6653_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_6654_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_6655_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_6656_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_6657_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_6658_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_6659_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_6660_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_6661_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_6662_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_6663_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_6664_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_6665_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_6666_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_6667_dvd__minus__add,axiom,
    ! [Q2: nat,N2: nat,R3: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N2 )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R3 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R3 @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_6668_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( ( ord_less_nat @ A @ D )
     => ( ( dvd_dvd_nat @ B @ D )
       => ( ord_less_eq_nat @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) @ ( minus_minus_nat @ D @ B ) ) ) ) ).

% diff_mod_le
thf(fact_6669_mod__nat__eqI,axiom,
    ! [R3: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R3 @ N2 )
     => ( ( ord_less_eq_nat @ R3 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R3 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R3 ) ) ) ) ).

% mod_nat_eqI
thf(fact_6670_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_6671_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_6672_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_6673_even__iff__mod__2__eq__zero,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
      = ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6674_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6675_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6676_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6677_even__iff__mod__2__eq__zero,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6678_odd__iff__mod__2__eq__one,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = one_on7727431528512463931l_num1 ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6679_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6680_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6681_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6682_odd__iff__mod__2__eq__one,axiom,
    ! [A: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6683_power__mono__odd,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_6684_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_6685_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_6686_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_6687_even__set__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ M @ A ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6688_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6689_even__set__bit__iff,axiom,
    ! [M: nat,A: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4894374433684937756l_num1 @ M @ A ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6690_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6691_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6692_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_6693_oddE,axiom,
    ! [A: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: uint32] :
            ( A
           != ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) @ one_one_uint32 ) ) ) ).

% oddE
thf(fact_6694_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_6695_oddE,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: word_N3645301735248828278l_num1] :
            ( A
           != ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B2 ) @ one_on7727431528512463931l_num1 ) ) ) ).

% oddE
thf(fact_6696_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_6697_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_6698_mod2__eq__if,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
          = zero_z3563351764282998399l_num1 ) )
      & ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
          = one_on7727431528512463931l_num1 ) ) ) ).

% mod2_eq_if
thf(fact_6699_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_6700_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_6701_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_6702_mod2__eq__if,axiom,
    ! [A: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = zero_zero_uint32 ) )
      & ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = one_one_uint32 ) ) ) ).

% mod2_eq_if
thf(fact_6703_parity__cases,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
         != zero_z3563351764282998399l_num1 ) )
     => ~ ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A )
         => ( ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
           != one_on7727431528512463931l_num1 ) ) ) ).

% parity_cases
thf(fact_6704_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_6705_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_6706_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_6707_parity__cases,axiom,
    ! [A: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
         != zero_zero_uint32 ) )
     => ~ ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
           != one_one_uint32 ) ) ) ).

% parity_cases
thf(fact_6708_zero__le__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6709_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6710_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6711_zero__le__odd__power,axiom,
    ! [N2: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6712_zero__le__odd__power,axiom,
    ! [N2: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6713_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6714_zero__le__even__power,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_6715_zero__le__even__power,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_6716_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_6717_divides__aux__def,axiom,
    ( unique5706413561485394159nteger
    = ( ^ [Qr: produc8923325533196201883nteger] :
          ( ( produc6174133586879617921nteger @ Qr )
          = zero_z3403309356797280102nteger ) ) ) ).

% divides_aux_def
thf(fact_6718_divides__aux__def,axiom,
    ( unique6322359934112328802ux_nat
    = ( ^ [Qr: product_prod_nat_nat] :
          ( ( product_snd_nat_nat @ Qr )
          = zero_zero_nat ) ) ) ).

% divides_aux_def
thf(fact_6719_divides__aux__def,axiom,
    ( unique6319869463603278526ux_int
    = ( ^ [Qr: product_prod_int_int] :
          ( ( product_snd_int_int @ Qr )
          = zero_zero_int ) ) ) ).

% divides_aux_def
thf(fact_6720_zero__less__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6721_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6722_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6723_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_6724_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_6725_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_6726_power__le__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6727_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6728_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6729_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_6730_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_6731_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_6732_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_6733_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_6734_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_6735_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_6736_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_6737_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_6738_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6739_even__mult__exp__div__exp__iff,axiom,
    ! [A: uint32,M: nat,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ ( times_times_uint32 @ A @ ( 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 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6740_even__mult__exp__div__exp__iff,axiom,
    ! [A: word_N3645301735248828278l_num1,M: nat,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ ( times_7065122842183080059l_num1 @ A @ ( 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 @ A @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6741_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6742_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6743_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_6744_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_6745_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_6746_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_6747_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_6748_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,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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_6749_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,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_6750_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_6751_unset__bit__Suc,axiom,
    ! [N2: nat,A: word_N3645301735248828278l_num1] :
      ( ( bit_se5331074070815623765l_num1 @ ( suc @ N2 ) @ A )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se5331074070815623765l_num1 @ N2 @ ( divide1791077408188789448l_num1 @ A @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6752_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6753_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6754_unset__bit__Suc,axiom,
    ! [N2: nat,A: uint32] :
      ( ( bit_se4315839071623982667uint32 @ ( suc @ N2 ) @ A )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ N2 @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6755_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6756_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_6757_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_6758_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
            @ ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6250501799366334488on_nat
                @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
                    @ ^ [Uu3: product_unit] :
                        ( produc2190226783763740553on_nat
                        @ ^ [Info4: option4927543243414619207at_nat] :
                            ( produc2606485630176857543on_nat
                            @ ^ [Deg5: nat] :
                                ( produc6124225815318652422on_nat
                                @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                    ( heap_T5999496708990702694on_nat
                                    @ ( refine_Imp_assert
                                      @ ( ( Info4 = Info3 )
                                        & ( Deg5 = Deg4 )
                                        & ( vEBT_is_Node @ T2 ) ) )
                                    @ ^ [Uv3: product_unit] :
                                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                        @ ^ [Mima2: product_prod_nat_nat] :
                                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                                              @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                @ ^ [L2: nat] :
                                                    ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                    @ ^ [H: nat] :
                                                        ( heap_T5999496708990702694on_nat
                                                        @ ( refine_Imp_assert
                                                          @ ( L2
                                                            = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                        @ ^ [Uw3: product_unit] :
                                                            ( heap_T5999496708990702694on_nat
                                                            @ ( refine_Imp_assert
                                                              @ ( H
                                                                = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                            @ ^ [Ux3: product_unit] :
                                                                ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                @ ^ [Uy3: product_unit] :
                                                                    ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                            @ ^ [Uz3: product_unit] :
                                                                                ( if_Hea5867803462524415986on_nat
                                                                                @ ( ( Minlow != none_nat )
                                                                                  & ( vEBT_VEBT_greater @ ( some_nat @ L2 ) @ Minlow ) )
                                                                                @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_predi3 ) @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                  @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                                @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_predi3 ) @ Summary4 @ Summary3 @ H )
                                                                                  @ ^ [Predsum: option_nat] :
                                                                                      ( heap_T5999496708990702694on_nat
                                                                                      @ ( refine_Imp_assert
                                                                                        @ ( ( Predsum = none_nat )
                                                                                          = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                            = none_nat ) ) )
                                                                                      @ ^ [Va4: product_unit] :
                                                                                          ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                          @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                        @ Info3 ) ) ) )
                        @ ( vEBT_c634343235235684882T_VEBT
                          @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                          @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                          @ T2 ) ) )
                @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( 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_6759_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_6760_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_6761_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 ) ) ) ) )
                          @ ^ [TreeList3: list_VEBT_VEBTi] :
                              ( heap_T7982501707604696571_VEBTi
                              @ ( refine_Imp_assert
                                @ ( ( size_s7982070591426661849_VEBTi @ TreeList3 )
                                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              @ ^ [Uu3: product_unit] :
                                  ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                                  @ ^ [Trees: array_VEBT_VEBTi] :
                                      ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
                    & ( ~ ( 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 ) ) ) ) )
                          @ ^ [TreeList3: list_VEBT_VEBTi] :
                              ( heap_T7982501707604696571_VEBTi
                              @ ( refine_Imp_assert
                                @ ( ( size_s7982070591426661849_VEBTi @ TreeList3 )
                                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                              @ ^ [Uu3: product_unit] :
                                  ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                                  @ ^ [Trees: array_VEBT_VEBTi] :
                                      ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.elims
thf(fact_6762_VEBT__internal_Ovebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V739175172307565963ildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList3: list_VEBT_VEBTi] :
                ( heap_T7982501707604696571_VEBTi
                @ ( refine_Imp_assert
                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ^ [Uu3: product_unit] :
                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                    @ ^ [Trees: array_VEBT_VEBTi] :
                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V739175172307565963ildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList3: list_VEBT_VEBTi] :
                ( heap_T7982501707604696571_VEBTi
                @ ( refine_Imp_assert
                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ^ [Uu3: product_unit] :
                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                    @ ^ [Trees: array_VEBT_VEBTi] :
                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(3)
thf(fact_6763_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 ) ) ) ) )
                          @ ^ [TreeList3: list_VEBT_VEBTi] :
                              ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                              @ ^ [Trees: array_VEBT_VEBTi] :
                                  ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) )
                    & ( ~ ( 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 ) ) ) ) )
                          @ ^ [TreeList3: list_VEBT_VEBTi] :
                              ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                              @ ^ [Trees: array_VEBT_VEBTi] :
                                  ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                  @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildupi.elims
thf(fact_6764_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6765_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: vEBT_VEBTi] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6766_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6767_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: vEBT_VEBTi] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6768_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: $o] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6769_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: $o] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6770_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6771_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( 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 @ B5 )
     => ( ! [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 @ ( B5 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6772_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_T8145700208782473153_VEBTi,C4: vEBT_VEBTi > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_T8145700208782473153_VEBTi] :
      ( ( comple2969382418784824877_VEBTi @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7173139186834293313_VEBTi @ B5 )
     => ( ! [Y3: vEBT_VEBTi] : ( 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_T1006145433769338483_VEBTi @ ( B5 @ F4 )
              @ ^ [Y: vEBT_VEBTi] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6773_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_T8145700208782473153_VEBTi,C4: vEBT_VEBTi > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_o] :
      ( ( comple2969382418784824877_VEBTi @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7173139186834293313_VEBTi @ B5 )
     => ( ! [Y3: vEBT_VEBTi] : ( 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_T3040810144269856602EBTi_o @ ( B5 @ F4 )
              @ ^ [Y: vEBT_VEBTi] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6774_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_inserti3: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] : ( P @ ( produc4062687810467144008_VEBTi @ ( produc2164094337957399884_VEBTi @ Vebt_inserti3 ) ) ) )
     => ( ( P
          @ ^ [Vebt_inserti3: vEBT_VEBT,T2: vEBT_VEBTi,Ti3: nat] :
              ( heap_T1489671670754571048_VEBTi
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A4: vEBT_VEBTi,B4: nat] :
                    ( vEBT_c6028912655521741485_VEBTi
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ B4 @ B4 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                        @ ^ [Minma: product_prod_nat_nat] :
                            ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                            @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                              @ ^ [Uu3: product_unit] :
                                  ( produc2624314226134418078_VEBTi
                                  @ ^ [Info4: option4927543243414619207at_nat] :
                                      ( produc7864503674192730076_VEBTi
                                      @ ^ [Deg5: nat] :
                                          ( produc9050507437146595227_VEBTi
                                          @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                              ( heap_T7982501707604696571_VEBTi
                                              @ ( refine_Imp_assert
                                                @ ( ( Info3 = Info4 )
                                                  & ( Deg4 = Deg5 ) ) )
                                              @ ^ [Uv3: 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 @ B4 @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ B4 ) )
                                                              @ ^ [Xn2: nat] :
                                                                  ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ B4 @ Mi3 ) @ ( heap_Time_return_nat @ B4 ) @ ( heap_Time_return_nat @ Mi3 ) )
                                                                  @ ^ [Minn: nat] :
                                                                      ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                      @ ^ [L2: nat] :
                                                                          ( heap_T7982501707604696571_VEBTi
                                                                          @ ( refine_Imp_assert
                                                                            @ ( L2
                                                                              = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ B4 @ Mi4 ) @ Mi4 @ B4 ) @ ( divide_divide_nat @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                          @ ^ [Uw3: product_unit] :
                                                                              ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                              @ ^ [H: nat] :
                                                                                  ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                                                  @ ^ [Len: nat] :
                                                                                      ( if_Hea8453224502484754311_VEBTi
                                                                                      @ ( ( ord_less_nat @ H @ Len )
                                                                                        & ~ ( ( B4 = Mi3 )
                                                                                            | ( B4 = Ma3 ) ) )
                                                                                      @ ( heap_T7982501707604696571_VEBTi
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( H
                                                                                            = ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ B4 @ Mi4 ) @ Mi4 @ B4 ) @ ( divide_divide_nat @ Deg5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                                        @ ^ [Ux3: product_unit] :
                                                                                            ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                                            @ ^ [Uy3: product_unit] :
                                                                                                ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                                                        @ ^ [Uz3: product_unit] :
                                                                                                            ( heap_T1006145433769338483_VEBTi @ ( F2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                            @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                                ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                                @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                                    ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( F2 @ Summary4 @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                                                                    @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                      @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                                  @ ( the_Pr8591224930841456533at_nat @ Info4 ) ) ) ) )
                                  @ ( vEBT_c634343235235684882T_VEBT
                                    @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                                    @ ^ [C3: $o,D2: $o] : undefi7074909574607331924T_VEBT
                                    @ X9 ) ) ) )
                        @ Info3 )
                    @ ^ [C3: $o,D2: $o] : ( if_Hea8453224502484754311_VEBTi @ ( B4 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ D2 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( B4 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ C3 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ C3 @ D2 ) ) ) )
                    @ A4 ) ) )
         => ( P @ vEBT_V3964819847710782039nserti ) ) ) ) ).

% VEBT_internal.vebt_inserti'.fixp_induct
thf(fact_6775_curryI,axiom,
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( produc1217013493180205570eger_o @ F @ A @ B ) ) ).

% curryI
thf(fact_6776_curryI,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( F @ ( produc5001842942810119800et_nat @ A @ B ) )
     => ( produc6216949301066131538_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_6777_curryI,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( F @ ( produc2245416461498447860et_nat @ A @ B ) )
     => ( produc5101573711933517782_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_6778_curryI,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( produc730925184835016917_int_o @ F @ A @ B ) ) ).

% curryI
thf(fact_6779_curryI,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( produc9098658269643458507_int_o @ F @ A @ B ) ) ).

% curryI
thf(fact_6780_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_6781_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_6782_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_6783_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_6784_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_6785_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_6786_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_6787_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_6788_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_6789_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_6790_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_6791_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_6792_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_6793_timeFrame_Ocases,axiom,
    ! [X2: produc2970534958541360184it_nat] :
      ( ! [N4: nat,R5: option_nat,H3: heap_e7401611519738050253t_unit,N7: nat] :
          ( X2
         != ( produc1430306010071928938it_nat @ N4 @ ( some_P2407035485129114418it_nat @ ( produc61566615109097733it_nat @ R5 @ ( produc584006145561248582it_nat @ H3 @ N7 ) ) ) ) )
     => ~ ! [N4: nat] :
            ( X2
           != ( produc1430306010071928938it_nat @ N4 @ none_P1551326421579882414it_nat ) ) ) ).

% timeFrame.cases
thf(fact_6794_timeFrame_Ocases,axiom,
    ! [X2: produc78474840892876564it_nat] :
      ( ! [N4: nat,R5: $o,H3: heap_e7401611519738050253t_unit,N7: nat] :
          ( X2
         != ( produc2174794033045126604it_nat @ N4 @ ( some_P468703482102919278it_nat @ ( produc6655106138504972685it_nat @ R5 @ ( produc584006145561248582it_nat @ H3 @ N7 ) ) ) ) )
     => ~ ! [N4: nat] :
            ( X2
           != ( produc2174794033045126604it_nat @ N4 @ none_P7668321371905463026it_nat ) ) ) ).

% timeFrame.cases
thf(fact_6795_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_6796_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_6797_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_6798_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_6799_curryE,axiom,
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F @ A @ B )
     => ( F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% curryE
thf(fact_6800_curryE,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F @ A @ B )
     => ( F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% curryE
thf(fact_6801_curryE,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F @ A @ B )
     => ( F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% curryE
thf(fact_6802_curryE,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_6803_curryE,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_6804_curryD,axiom,
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F @ A @ B )
     => ( F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% curryD
thf(fact_6805_curryD,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F @ A @ B )
     => ( F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% curryD
thf(fact_6806_curryD,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F @ A @ B )
     => ( F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% curryD
thf(fact_6807_curryD,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_6808_curryD,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_6809_TBOUND__empty,axiom,
    ! [T: nat] :
      ( time_TBOUND_nat
      @ ( heap_Time_Heap_nat2
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P281974696781278558it_nat )
      @ T ) ).

% TBOUND_empty
thf(fact_6810_TBOUND__empty,axiom,
    ! [T: nat] :
      ( time_T5737551269749752165_VEBTi
      @ ( heap_T1489671670754571048_VEBTi
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat )
      @ T ) ).

% TBOUND_empty
thf(fact_6811_TBOUND__empty,axiom,
    ! [T: nat] :
      ( time_T8353473612707095248on_nat
      @ ( heap_T5286843759275942675on_nat
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat )
      @ T ) ).

% TBOUND_empty
thf(fact_6812_TBOUND__empty,axiom,
    ! [T: nat] :
      ( time_TBOUND_o
      @ ( heap_Time_Heap_o2
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat )
      @ T ) ).

% TBOUND_empty
thf(fact_6813_refines__empty,axiom,
    ! [M: heap_T8145700208782473153_VEBTi] :
      ( refine5565527176597971370_VEBTi @ M
      @ ( heap_T1489671670754571048_VEBTi
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat ) ) ).

% refines_empty
thf(fact_6814_refines__empty,axiom,
    ! [M: heap_T2636463487746394924on_nat] :
      ( refine7594492741263601813on_nat @ M
      @ ( heap_T5286843759275942675on_nat
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) ) ).

% refines_empty
thf(fact_6815_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_6816_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_6817_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_6818_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_6819_zdvd__reduce,axiom,
    ! [K: int,N2: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_6820_zdvd__period,axiom,
    ! [A: int,D: int,X2: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X2 @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X2 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_6821_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_6822_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_6823_fold__if__return,axiom,
    ! [B: $o,C: option_nat,D: option_nat] :
      ( ( B
       => ( ( heap_T3487192422709364219on_nat @ C )
          = ( heap_T3487192422709364219on_nat @ ( if_option_nat @ B @ C @ D ) ) ) )
      & ( ~ B
       => ( ( heap_T3487192422709364219on_nat @ D )
          = ( heap_T3487192422709364219on_nat @ ( if_option_nat @ B @ C @ D ) ) ) ) ) ).

% fold_if_return
thf(fact_6824_fold__if__return,axiom,
    ! [B: $o,C: nat,D: nat] :
      ( ( B
       => ( ( heap_Time_return_nat @ C )
          = ( heap_Time_return_nat @ ( if_nat @ B @ C @ D ) ) ) )
      & ( ~ B
       => ( ( heap_Time_return_nat @ D )
          = ( heap_Time_return_nat @ ( if_nat @ B @ C @ D ) ) ) ) ) ).

% fold_if_return
thf(fact_6825_fold__if__return,axiom,
    ! [B: $o,C: $o,D: $o] :
      ( ( B
       => ( ( heap_Time_return_o @ C )
          = ( heap_Time_return_o
            @ ( ( B
               => C )
              & ( ~ B
               => D ) ) ) ) )
      & ( ~ B
       => ( ( heap_Time_return_o @ D )
          = ( heap_Time_return_o
            @ ( ( B
               => C )
              & ( ~ B
               => D ) ) ) ) ) ) ).

% fold_if_return
thf(fact_6826_fold__if__return,axiom,
    ! [B: $o,C: vEBT_VEBTi,D: vEBT_VEBTi] :
      ( ( B
       => ( ( heap_T3630416162098727440_VEBTi @ C )
          = ( heap_T3630416162098727440_VEBTi @ ( if_VEBT_VEBTi @ B @ C @ D ) ) ) )
      & ( ~ B
       => ( ( heap_T3630416162098727440_VEBTi @ D )
          = ( heap_T3630416162098727440_VEBTi @ ( if_VEBT_VEBTi @ B @ C @ D ) ) ) ) ) ).

% fold_if_return
thf(fact_6827_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_6828_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_6829_aset_I10_J,axiom,
    ! [D: int,D5: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X6 @ D5 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_6830_aset_I9_J,axiom,
    ! [D: int,D5: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X6 @ D5 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_6831_bset_I10_J,axiom,
    ! [D: int,D5: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ D5 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_6832_bset_I9_J,axiom,
    ! [D: int,D5: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D5 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ D5 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_6833_distrib__if__bind,axiom,
    ! [B: $o,C: heap_T4980287057938770641_VEBTi,D: heap_T4980287057938770641_VEBTi,F: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( B
       => ( ( heap_T5877712393672139267_VEBTi @ ( if_Hea811341299636385687_VEBTi @ B @ C @ D ) @ F )
          = ( heap_T5877712393672139267_VEBTi @ C @ F ) ) )
      & ( ~ B
       => ( ( heap_T5877712393672139267_VEBTi @ ( if_Hea811341299636385687_VEBTi @ B @ C @ D ) @ F )
          = ( heap_T5877712393672139267_VEBTi @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6834_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_o,D: heap_Time_Heap_o,F: $o > heap_T8145700208782473153_VEBTi] :
      ( ( B
       => ( ( heap_T5998771940306268294_VEBTi @ ( if_Heap_Time_Heap_o @ B @ C @ D ) @ F )
          = ( heap_T5998771940306268294_VEBTi @ C @ F ) ) )
      & ( ~ B
       => ( ( heap_T5998771940306268294_VEBTi @ ( if_Heap_Time_Heap_o @ B @ C @ D ) @ F )
          = ( heap_T5998771940306268294_VEBTi @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6835_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: nat > heap_T8145700208782473153_VEBTi] :
      ( ( B
       => ( ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ B @ C @ D ) @ F )
          = ( heap_T844888390831797134_VEBTi @ C @ F ) ) )
      & ( ~ B
       => ( ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ B @ C @ D ) @ F )
          = ( heap_T844888390831797134_VEBTi @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6836_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: nat > heap_T2636463487746394924on_nat] :
      ( ( B
       => ( ( heap_T8222160169144143993on_nat @ ( if_Hea2662716070787841314ap_nat @ B @ C @ D ) @ F )
          = ( heap_T8222160169144143993on_nat @ C @ F ) ) )
      & ( ~ B
       => ( ( heap_T8222160169144143993on_nat @ ( if_Hea2662716070787841314ap_nat @ B @ C @ D ) @ F )
          = ( heap_T8222160169144143993on_nat @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6837_distrib__if__bind,axiom,
    ! [B: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: nat > heap_Time_Heap_o] :
      ( ( B
       => ( ( heap_Time_bind_nat_o @ ( if_Hea2662716070787841314ap_nat @ B @ C @ D ) @ F )
          = ( heap_Time_bind_nat_o @ C @ F ) ) )
      & ( ~ B
       => ( ( heap_Time_bind_nat_o @ ( if_Hea2662716070787841314ap_nat @ B @ C @ D ) @ F )
          = ( heap_Time_bind_nat_o @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6838_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_diff_iff
thf(fact_6839_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_6840_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_6841_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_6842_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_6843_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_6844_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
          @ ^ [T2: vEBT_VEBTi,X: nat] :
              ( vEBT_c6104975476656191286Heap_o
              @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeList3: array_VEBT_VEBTi,Summary3: 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 @ Deg4 @ 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [H: nat] :
                                      ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [L2: nat] :
                                          ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList3 )
                                          @ ^ [Len: nat] :
                                              ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                              @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList3 @ H )
                                                @ ^ [Th: vEBT_VEBTi] : ( produc5685940877448195828Heap_o @ Vebt_memberi3 @ Th @ L2 ) )
                                              @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A4: $o,B4: $o] :
                  ( heap_Time_return_o
                  @ ( ( ( X = zero_zero_nat )
                     => A4 )
                    & ( ( X != zero_zero_nat )
                     => ( ( ( X = one_one_nat )
                         => B4 )
                        & ( X = one_one_nat ) ) ) ) )
              @ T2 )
          @ X2 ) ) ).

% vebt_memberi.mono
thf(fact_6845_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_6846_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_6847_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_6848_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_6849_dvd__productE,axiom,
    ! [P4: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P4
              = ( times_times_nat @ X3 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_6850_dvd__productE,axiom,
    ! [P4: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P4
              = ( times_times_int @ X3 @ Y3 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_6851_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B8: nat,C6: nat] :
          ( ( A
            = ( times_times_nat @ B8 @ C6 ) )
          & ( dvd_dvd_nat @ B8 @ B )
          & ( dvd_dvd_nat @ C6 @ C ) ) ) ).

% division_decomp
thf(fact_6852_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B8: int,C6: int] :
          ( ( A
            = ( times_times_int @ B8 @ C6 ) )
          & ( dvd_dvd_int @ B8 @ B )
          & ( dvd_dvd_int @ C6 @ C ) ) ) ).

% division_decomp
thf(fact_6853_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( P @ A3 @ B2 )
          = ( P @ B2 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B2: nat] :
              ( ( P @ A3 @ B2 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_6854_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_6855_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_6856_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_6857_of__list__rule,axiom,
    ! [Xs2: list_VEBT_VEBTi] :
      ( hoare_3353465787467722821_VEBTi @ one_one_assn @ ( array_615059503499738533_VEBTi @ Xs2 )
      @ ^ [R: array_VEBT_VEBTi] : ( snga_assn_VEBT_VEBTi @ R @ Xs2 ) ) ).

% of_list_rule
thf(fact_6858_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_6859_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_6860_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_6861_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_6862_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_6863_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_6864_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
     => ( ! [M2: nat,N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ( P @ N4 @ ( modulo_modulo_nat @ M2 @ N4 ) )
             => ( P @ M2 @ N4 ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_6865_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
            @ ^ [T2: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6104975476656191286Heap_o
                @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T2 ) )
                    @ ^ [Uu3: 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 @ Deg4 @ 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
                                        @ ^ [Info4: option4927543243414619207at_nat] :
                                            ( produc5946672270950774087Heap_o
                                            @ ^ [Deg5: nat] :
                                                ( produc5048428016959714504Heap_o
                                                @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                                    ( heap_T7577940988442287570unit_o
                                                    @ ( refine_Imp_assert
                                                      @ ( ( Info3 = Info4 )
                                                        & ( Deg4 = Deg5 ) ) )
                                                    @ ^ [Uv3: product_unit] :
                                                        ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                            @ ^ [L2: nat] :
                                                                ( heap_T7577940988442287570unit_o
                                                                @ ( refine_Imp_assert
                                                                  @ ( ( L2
                                                                      = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                    & ( H
                                                                      = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                                @ ^ [Uw3: product_unit] :
                                                                    ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                                    @ ^ [Len: nat] :
                                                                        ( heap_T7577940988442287570unit_o
                                                                        @ ( refine_Imp_assert
                                                                          @ ( Len
                                                                            = ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                        @ ^ [Ux3: 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                                  & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                                                                              @ ^ [Uy3: product_unit] :
                                                                                  ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                                                  @ ^ [Th: vEBT_VEBTi] : ( produc2663629013181010545Heap_o @ ( produc8381543706267210711Heap_o @ Vebt_memberi4 ) @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                            @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                        @ ( vEBT_c634343235235684882T_VEBT
                                          @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                                          @ ^ [A4: $o,B4: $o] : undefi7074909574607331924T_VEBT
                                          @ T2 ) ) ) ) ) ) ) )
                        @ Info3 ) )
                @ ^ [A4: $o,B4: $o] :
                    ( heap_Time_return_o
                    @ ( ( ( X = zero_zero_nat )
                       => A4 )
                      & ( ( X != zero_zero_nat )
                       => ( ( ( X = one_one_nat )
                           => B4 )
                          & ( X = one_one_nat ) ) ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_memberi'.mono
thf(fact_6866_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X2: nat,Y2: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X2 )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
            | ( ( times_times_nat @ B @ X2 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D ) ) )
         => ? [X3: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_6867_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_6868_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_6869_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,X2: option_nat] :
      ( ( H2 = H4 )
     => ( heap_T306965388786959644on_nat @ ( heap_T3487192422709364219on_nat @ X2 ) @ H2 @ H4 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6870_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,X2: nat] :
      ( ( H2 = H4 )
     => ( heap_Time_effect_nat @ ( heap_Time_return_nat @ X2 ) @ H2 @ H4 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6871_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,X2: $o] :
      ( ( H2 = H4 )
     => ( heap_Time_effect_o @ ( heap_Time_return_o @ X2 ) @ H2 @ H4 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6872_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,X2: vEBT_VEBTi] :
      ( ( H2 = H4 )
     => ( heap_T2071195472996403633_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ H2 @ H4 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6873_effect__returnE,axiom,
    ! [X2: option_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T3487192422709364219on_nat @ X2 ) @ H2 @ H4 @ R3 @ N2 )
     => ~ ( ( R3 = X2 )
         => ( ( H4 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6874_effect__returnE,axiom,
    ! [X2: nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: nat,N2: nat] :
      ( ( heap_Time_effect_nat @ ( heap_Time_return_nat @ X2 ) @ H2 @ H4 @ R3 @ N2 )
     => ~ ( ( R3 = X2 )
         => ( ( H4 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6875_effect__returnE,axiom,
    ! [X2: $o,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_Time_return_o @ X2 ) @ H2 @ H4 @ R3 @ N2 )
     => ~ ( ( R3 = X2 )
         => ( ( H4 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6876_effect__returnE,axiom,
    ! [X2: vEBT_VEBTi,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ H2 @ H4 @ R3 @ N2 )
     => ~ ( ( R3 = X2 )
         => ( ( H4 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6877_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T844888390831797134_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: nat,N1: nat] :
            ( ( heap_Time_effect_nat @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6878_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: vEBT_VEBTi > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T1006145433769338483_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N1: nat] :
            ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6879_effect__bindE,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T5998771940306268294_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: $o,N1: nat] :
            ( ( heap_Time_effect_o @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6880_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_Time_bind_nat_o @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: nat,N1: nat] :
            ( ( heap_Time_effect_nat @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect_o @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6881_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: vEBT_VEBTi > heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_T3040810144269856602EBTi_o @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N1: nat] :
            ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect_o @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6882_effect__bindE,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_Time_bind_o_o @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: $o,N1: nat] :
            ( ( heap_Time_effect_o @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect_o @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6883_effect__bindE,axiom,
    ! [F: heap_T4980287057938770641_VEBTi,G: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T5877712393672139267_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: list_VEBT_VEBTi,N1: nat] :
            ( ( heap_T33481931004607297_VEBTi @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6884_effect__bindE,axiom,
    ! [F: heap_T2636463487746394924on_nat,G: option_nat > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T5661892481228163294_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: option_nat,N1: nat] :
            ( ( heap_T306965388786959644on_nat @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6885_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T8222160169144143993on_nat @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: nat,N1: nat] :
            ( ( heap_Time_effect_nat @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T306965388786959644on_nat @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6886_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: vEBT_VEBTi > heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H8: heap_e7401611519738050253t_unit,R4: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T2868974464944644318on_nat @ F @ G ) @ H2 @ H8 @ R4 @ N2 )
     => ~ ! [H5: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N1: nat] :
            ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H5 @ R5 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T306965388786959644on_nat @ ( G @ R5 ) @ H5 @ H8 @ R4 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6887_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: nat,N2: nat,G: nat > heap_T8145700208782473153_VEBTi,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N5: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T844888390831797134_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6888_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: nat,N2: nat,G: nat > heap_Time_Heap_o,H8: heap_e7401611519738050253t_unit,R4: $o,N5: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_Time_effect_o @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_Time_effect_o @ ( heap_Time_bind_nat_o @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6889_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N2: nat,G: vEBT_VEBTi > heap_T8145700208782473153_VEBTi,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N5: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T1006145433769338483_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6890_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N2: nat,G: vEBT_VEBTi > heap_Time_Heap_o,H8: heap_e7401611519738050253t_unit,R4: $o,N5: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_Time_effect_o @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_Time_effect_o @ ( heap_T3040810144269856602EBTi_o @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6891_effect__bindI,axiom,
    ! [F: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: $o,N2: nat,G: $o > heap_T8145700208782473153_VEBTi,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N5: nat] :
      ( ( heap_Time_effect_o @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T5998771940306268294_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6892_effect__bindI,axiom,
    ! [F: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: $o,N2: nat,G: $o > heap_Time_Heap_o,H8: heap_e7401611519738050253t_unit,R4: $o,N5: nat] :
      ( ( heap_Time_effect_o @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_Time_effect_o @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_Time_effect_o @ ( heap_Time_bind_o_o @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6893_effect__bindI,axiom,
    ! [F: heap_T4980287057938770641_VEBTi,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: list_VEBT_VEBTi,N2: nat,G: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N5: nat] :
      ( ( heap_T33481931004607297_VEBTi @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T5877712393672139267_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6894_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: nat,N2: nat,G: nat > heap_T2636463487746394924on_nat,H8: heap_e7401611519738050253t_unit,R4: option_nat,N5: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_T306965388786959644on_nat @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_T306965388786959644on_nat @ ( heap_T8222160169144143993on_nat @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6895_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N2: nat,G: vEBT_VEBTi > heap_T2636463487746394924on_nat,H8: heap_e7401611519738050253t_unit,R4: option_nat,N5: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_T306965388786959644on_nat @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_T306965388786959644on_nat @ ( heap_T2868974464944644318on_nat @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6896_effect__bindI,axiom,
    ! [F: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: option_nat,N2: nat,G: option_nat > heap_T8145700208782473153_VEBTi,H8: heap_e7401611519738050253t_unit,R4: vEBT_VEBTi,N5: nat] :
      ( ( heap_T306965388786959644on_nat @ F @ H2 @ H4 @ R3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H4 @ H8 @ R4 @ N5 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T5661892481228163294_VEBTi @ F @ G ) @ H2 @ H8 @ R4 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_6897_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
          @ ^ [T2: vEBT_VEBTi,X: nat] :
              ( vEBT_c6250501799366334488on_nat
              @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ^ [Mima2: product_prod_nat_nat] :
                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                        @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                          @ ^ [L2: nat] :
                              ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                              @ ^ [H: nat] :
                                  ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Minlow ) )
                                          @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_predi4 @ Aktnode @ L2 )
                                            @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                          @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_predi4 @ Summary3 @ H )
                                            @ ^ [Predsum: option_nat] :
                                                ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
              @ T2 )
          @ X2 ) ) ).

% vebt_predi.mono
thf(fact_6898_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
          @ ^ [T2: vEBT_VEBTi,X: nat] :
              ( vEBT_c6250501799366334488on_nat
              @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ^ [Mima2: product_prod_nat_nat] :
                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                        @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                          @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                            @ ^ [L2: nat] :
                                ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                @ ^ [H: nat] :
                                    ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Maxlow ) )
                                            @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_succi4 @ Aktnode @ L2 )
                                              @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                            @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_succi4 @ Summary3 @ H )
                                              @ ^ [Succsum: option_nat] :
                                                  ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                  @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A4: $o,B4: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
              @ T2 )
          @ X2 ) ) ).

% vebt_succi.mono
thf(fact_6899_vebt__inserti_Omono,axiom,
    ! [X2: produc3881548065746020326Ti_nat] :
      ( comple2284608890766496472_VEBTi @ ( partia6690842624828592406Ti_nat @ heap_T7173139186834293313_VEBTi ) @ heap_T7173139186834293313_VEBTi
      @ ^ [Vebt_inserti4: produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi] :
          ( produc3255295512018472142_VEBTi
          @ ^ [T2: vEBT_VEBTi,X: nat] :
              ( vEBT_c6028912655521741485_VEBTi
              @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                  ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                  @ ^ [Minma: product_prod_nat_nat] :
                      ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                      @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [L2: nat] :
                                            ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [H: nat] :
                                                ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                @ ^ [Len: nat] :
                                                    ( if_Hea8453224502484754311_VEBTi
                                                    @ ( ( ord_less_nat @ H @ Len )
                                                      & ~ ( ( X = Mi3 )
                                                          | ( X = Ma3 ) ) )
                                                    @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                      @ ^ [Node: vEBT_VEBTi] :
                                                          ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                          @ ^ [Empt: $o] :
                                                              ( heap_T1006145433769338483_VEBTi @ ( produc5159149307777246319_VEBTi @ Vebt_inserti4 @ Node @ L2 )
                                                              @ ^ [Newnode2: vEBT_VEBTi] :
                                                                  ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                  @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                      ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( produc5159149307777246319_VEBTi @ Vebt_inserti4 @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                      @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                    @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A4: $o,B4: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B4 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
              @ T2 )
          @ X2 ) ) ).

% vebt_inserti.mono
thf(fact_6900_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_6901_vebt__buildupi_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList3: list_VEBT_VEBTi] :
                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                @ ^ [Trees: array_VEBT_VEBTi] :
                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList3: list_VEBT_VEBTi] :
                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                @ ^ [Trees: array_VEBT_VEBTi] :
                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ).

% vebt_buildupi.simps(3)
thf(fact_6902_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_inserti4: produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi] : ( P @ ( produc5159149307777246319_VEBTi @ Vebt_inserti4 ) ) )
     => ( ( P
          @ ^ [Vebt_inserti4: vEBT_VEBTi,T2: nat] :
              ( heap_T1489671670754571048_VEBTi
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat ) )
       => ( ! [F2: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A4: nat] :
                    ( vEBT_c6028912655521741485_VEBTi
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                        @ ^ [Minma: product_prod_nat_nat] :
                            ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                            @ ( 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 @ A4 @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ A4 ) )
                                      @ ^ [Xn2: nat] :
                                          ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ A4 @ Mi3 ) @ ( heap_Time_return_nat @ A4 ) @ ( heap_Time_return_nat @ Mi3 ) )
                                          @ ^ [Minn: nat] :
                                              ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                              @ ^ [L2: nat] :
                                                  ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [H: nat] :
                                                      ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                      @ ^ [Len: nat] :
                                                          ( if_Hea8453224502484754311_VEBTi
                                                          @ ( ( ord_less_nat @ H @ Len )
                                                            & ~ ( ( A4 = Mi3 )
                                                                | ( A4 = Ma3 ) ) )
                                                          @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                            @ ^ [Node: vEBT_VEBTi] :
                                                                ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                @ ^ [Empt: $o] :
                                                                    ( heap_T1006145433769338483_VEBTi @ ( F2 @ Node @ L2 )
                                                                    @ ^ [Newnode2: vEBT_VEBTi] :
                                                                        ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                        @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                            ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( F2 @ Summary3 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary3 ) )
                                                                            @ ^ [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 ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                          @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B4: $o,C3: $o] : ( if_Hea8453224502484754311_VEBTi @ ( A4 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ C3 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( A4 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ B4 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ B4 @ C3 ) ) ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_inserti ) ) ) ) ).

% vebt_inserti.fixp_induct
thf(fact_6903_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_6904_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,T2: vEBT_VEBTi,Ti3: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A4: vEBT_VEBTi,B4: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                        @ ^ [Uu3: product_unit] :
                            ( produc2190226783763740553on_nat
                            @ ^ [Info4: option4927543243414619207at_nat] :
                                ( produc2606485630176857543on_nat
                                @ ^ [Deg5: nat] :
                                    ( produc6124225815318652422on_nat
                                    @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                        ( heap_T5999496708990702694on_nat
                                        @ ( refine_Imp_assert
                                          @ ( ( Info4 = Info3 )
                                            & ( Deg5 = Deg4 )
                                            & ( vEBT_is_Node @ X9 ) ) )
                                        @ ^ [Uv3: product_unit] :
                                            ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ^ [Mima2: product_prod_nat_nat] :
                                                ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ B4 @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                                                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ B4 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                    @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [L2: nat] :
                                                          ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                          @ ^ [H: nat] :
                                                              ( heap_T5999496708990702694on_nat
                                                              @ ( refine_Imp_assert
                                                                @ ( L2
                                                                  = ( vEBT_VEBT_low @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              @ ^ [Uw3: product_unit] :
                                                                  ( heap_T5999496708990702694on_nat
                                                                  @ ( refine_Imp_assert
                                                                    @ ( H
                                                                      = ( vEBT_VEBT_high @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                  @ ^ [Ux3: product_unit] :
                                                                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                      @ ^ [Uy3: product_unit] :
                                                                          ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                                  @ ^ [Uz3: product_unit] :
                                                                                      ( if_Hea5867803462524415986on_nat
                                                                                      @ ( ( Maxlow != none_nat )
                                                                                        & ( vEBT_VEBT_less @ ( some_nat @ L2 ) @ Maxlow ) )
                                                                                      @ ( heap_T3669509953089699273on_nat @ ( F2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                        @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                                      @ ( heap_T3669509953089699273on_nat @ ( F2 @ Summary4 @ Summary3 @ H )
                                                                                        @ ^ [Succsum: option_nat] :
                                                                                            ( heap_T5999496708990702694on_nat
                                                                                            @ ( refine_Imp_assert
                                                                                              @ ( ( Succsum = none_nat )
                                                                                                = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                                  = none_nat ) ) )
                                                                                            @ ^ [Va4: product_unit] :
                                                                                                ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                                @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                            @ Info3 ) ) ) )
                            @ ( vEBT_c634343235235684882T_VEBT
                              @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                              @ ^ [C3: $o,D2: $o] : undefi7074909574607331924T_VEBT
                              @ X9 ) ) )
                    @ ^ [C3: $o,D2: $o] : ( if_Hea5867803462524415986on_nat @ ( B4 = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ D2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                    @ A4 ) ) )
         => ( P @ vEBT_VEBT_vebt_succi ) ) ) ) ).

% VEBT_internal.vebt_succi'.fixp_induct
thf(fact_6905_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,T2: vEBT_VEBTi,Ti3: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A4: vEBT_VEBTi,B4: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                        @ ^ [Uu3: product_unit] :
                            ( produc2190226783763740553on_nat
                            @ ^ [Info4: option4927543243414619207at_nat] :
                                ( produc2606485630176857543on_nat
                                @ ^ [Deg5: nat] :
                                    ( produc6124225815318652422on_nat
                                    @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                        ( heap_T5999496708990702694on_nat
                                        @ ( refine_Imp_assert
                                          @ ( ( Info4 = Info3 )
                                            & ( Deg5 = Deg4 )
                                            & ( vEBT_is_Node @ X9 ) ) )
                                        @ ^ [Uv3: product_unit] :
                                            ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ^ [Mima2: product_prod_nat_nat] :
                                                ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ B4 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                                                  @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                    @ ^ [L2: nat] :
                                                        ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_T5999496708990702694on_nat
                                                            @ ( refine_Imp_assert
                                                              @ ( L2
                                                                = ( vEBT_VEBT_low @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                            @ ^ [Uw3: product_unit] :
                                                                ( heap_T5999496708990702694on_nat
                                                                @ ( refine_Imp_assert
                                                                  @ ( H
                                                                    = ( vEBT_VEBT_high @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                @ ^ [Ux3: product_unit] :
                                                                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                    @ ^ [Uy3: product_unit] :
                                                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ TreeList3 @ H ) ) ) )
                                                                                @ ^ [Uz3: product_unit] :
                                                                                    ( if_Hea5867803462524415986on_nat
                                                                                    @ ( ( Minlow != none_nat )
                                                                                      & ( vEBT_VEBT_greater @ ( some_nat @ L2 ) @ Minlow ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( F2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                      @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( F2 @ Summary4 @ Summary3 @ H )
                                                                                      @ ^ [Predsum: option_nat] :
                                                                                          ( heap_T5999496708990702694on_nat
                                                                                          @ ( refine_Imp_assert
                                                                                            @ ( ( Predsum = none_nat )
                                                                                              = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                                = none_nat ) ) )
                                                                                          @ ^ [Va4: product_unit] :
                                                                                              ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ B4 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                              @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                            @ Info3 ) ) ) )
                            @ ( vEBT_c634343235235684882T_VEBT
                              @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                              @ ^ [C3: $o,D2: $o] : undefi7074909574607331924T_VEBT
                              @ X9 ) ) )
                    @ ^ [C3: $o,D2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ ( if_Hea5867803462524415986on_nat @ D2 @ ( 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 @ ( B4 = 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 ) ) )
                    @ A4 ) ) )
         => ( P @ vEBT_VEBT_vebt_predi ) ) ) ) ).

% VEBT_internal.vebt_predi'.fixp_induct
thf(fact_6906_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 ) ) ) ) )
                            @ ^ [TreeList3: list_VEBT_VEBTi] :
                                ( heap_T7982501707604696571_VEBTi
                                @ ( refine_Imp_assert
                                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList3 )
                                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                @ ^ [Uu3: product_unit] :
                                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                                    @ ^ [Trees: array_VEBT_VEBTi] :
                                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
                      & ( ~ ( 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 ) ) ) ) )
                            @ ^ [TreeList3: list_VEBT_VEBTi] :
                                ( heap_T7982501707604696571_VEBTi
                                @ ( refine_Imp_assert
                                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList3 )
                                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                @ ^ [Uu3: product_unit] :
                                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                                    @ ^ [Trees: array_VEBT_VEBTi] :
                                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V254170901696579886pi_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.pelims
thf(fact_6907_prod__decode__aux_Ocases,axiom,
    ! [X2: product_prod_nat_nat] :
      ~ ! [K3: nat,M2: nat] :
          ( X2
         != ( product_Pair_nat_nat @ K3 @ M2 ) ) ).

% prod_decode_aux.cases
thf(fact_6908_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ~ ! [N4: nat] :
            ( X2
           != ( suc @ N4 ) ) ) ).

% list_decode.cases
thf(fact_6909_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 ) ) ) ) )
                            @ ^ [TreeList3: list_VEBT_VEBTi] :
                                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                                @ ^ [Trees: array_VEBT_VEBTi] :
                                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) )
                      & ( ~ ( 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 ) ) ) ) )
                            @ ^ [TreeList3: list_VEBT_VEBTi] :
                                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList3 )
                                @ ^ [Trees: array_VEBT_VEBTi] :
                                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v1230518104690509829pi_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildupi.pelims
thf(fact_6910_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_6911_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_6912_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_6913_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_6914_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_6915_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_6916_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,T2: vEBT_VEBTi,Ti3: nat] :
              ( heap_Time_Heap_o2
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A4: vEBT_VEBTi,B4: nat] :
                    ( vEBT_c6104975476656191286Heap_o
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                        @ ^ [Uu3: 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 @ Deg4 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                                  @ ( if_Heap_Time_Heap_o @ ( B4 = Mi3 ) @ ( heap_Time_return_o @ $true )
                                    @ ( if_Heap_Time_Heap_o @ ( B4 = Ma3 ) @ ( heap_Time_return_o @ $true )
                                      @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ B4 @ Mi3 ) @ ( heap_Time_return_o @ $false )
                                        @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ B4 ) @ ( heap_Time_return_o @ $false )
                                          @ ( produc1330493526443650053Heap_o
                                            @ ^ [Info4: option4927543243414619207at_nat] :
                                                ( produc5946672270950774087Heap_o
                                                @ ^ [Deg5: nat] :
                                                    ( produc5048428016959714504Heap_o
                                                    @ ^ [TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
                                                        ( heap_T7577940988442287570unit_o
                                                        @ ( refine_Imp_assert
                                                          @ ( ( Info3 = Info4 )
                                                            & ( Deg4 = Deg5 ) ) )
                                                        @ ^ [Uv3: product_unit] :
                                                            ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                            @ ^ [H: nat] :
                                                                ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                @ ^ [L2: nat] :
                                                                    ( heap_T7577940988442287570unit_o
                                                                    @ ( refine_Imp_assert
                                                                      @ ( ( L2
                                                                          = ( vEBT_VEBT_low @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                        & ( H
                                                                          = ( vEBT_VEBT_high @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                                    @ ^ [Uw3: product_unit] :
                                                                        ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray )
                                                                        @ ^ [Len: nat] :
                                                                            ( heap_T7577940988442287570unit_o
                                                                            @ ( refine_Imp_assert
                                                                              @ ( Len
                                                                                = ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) )
                                                                            @ ^ [Ux3: product_unit] :
                                                                                ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                                                @ ( heap_T7577940988442287570unit_o
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( ( H
                                                                                        = ( vEBT_VEBT_high @ B4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                                      & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                                                                                  @ ^ [Uy3: product_unit] :
                                                                                      ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray @ H )
                                                                                      @ ^ [Th: vEBT_VEBTi] : ( F2 @ ( nth_VEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                                @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                            @ ( vEBT_c634343235235684882T_VEBT
                                              @ ^ [Info4: option4927543243414619207at_nat,Deg5: nat,TreeList3: list_VEBT_VEBT,Summary4: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info4 @ ( produc1750349459881913976T_VEBT @ Deg5 @ ( produc6691630295680060561T_VEBT @ TreeList3 @ Summary4 ) ) )
                                              @ ^ [C3: $o,D2: $o] : undefi7074909574607331924T_VEBT
                                              @ X9 ) ) ) ) ) ) ) )
                            @ Info3 ) )
                    @ ^ [C3: $o,D2: $o] :
                        ( heap_Time_return_o
                        @ ( ( ( B4 = zero_zero_nat )
                           => C3 )
                          & ( ( B4 != zero_zero_nat )
                           => ( ( ( B4 = one_one_nat )
                               => D2 )
                              & ( B4 = one_one_nat ) ) ) ) )
                    @ A4 ) ) )
         => ( P @ vEBT_V854960066525838166emberi ) ) ) ) ).

% VEBT_internal.vebt_memberi'.fixp_induct
thf(fact_6917_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,T2: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F2: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A4: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                        @ ^ [Mima2: product_prod_nat_nat] :
                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ A4 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                              @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ A4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                @ ^ [L2: nat] :
                                    ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ A4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [H: nat] :
                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Minlow ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( F2 @ Aktnode @ L2 )
                                                  @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( F2 @ Summary3 @ H )
                                                  @ ^ [Predsum: option_nat] :
                                                      ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ A4 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                      @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B4: $o,C3: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) @ ( if_Hea5867803462524415986on_nat @ C3 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( A4 = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ B4 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_predi ) ) ) ) ).

% vebt_predi.fixp_induct
thf(fact_6918_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,T2: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F2: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A4: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeArray: array_VEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                        @ ^ [Mima2: product_prod_nat_nat] :
                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg4 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ A4 @ ( product_fst_nat_nat @ Mima2 ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) )
                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima2 ) @ A4 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ A4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [L2: nat] :
                                      ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ A4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [H: nat] :
                                          ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ 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 @ L2 ) @ Maxlow ) )
                                                  @ ( heap_T3669509953089699273on_nat @ ( F2 @ Aktnode @ L2 )
                                                    @ ^ [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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                  @ ( heap_T3669509953089699273on_nat @ ( F2 @ Summary3 @ H )
                                                    @ ^ [Succsum: option_nat] :
                                                        ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                        @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B4: $o,C3: $o] : ( if_Hea5867803462524415986on_nat @ ( A4 = 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 ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_succi ) ) ) ) ).

% vebt_succi.fixp_induct
thf(fact_6919_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,T2: nat] :
              ( heap_Time_Heap_o2
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat ) )
       => ( ! [F2: vEBT_VEBTi > nat > heap_Time_Heap_o] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A4: nat] :
                    ( vEBT_c6104975476656191286Heap_o
                    @ ^ [Info3: option4927543243414619207at_nat,Deg4: nat,TreeList3: array_VEBT_VEBTi,Summary3: 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 @ Deg4 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                              @ ( if_Heap_Time_Heap_o @ ( A4 = Mi3 ) @ ( heap_Time_return_o @ $true )
                                @ ( if_Heap_Time_Heap_o @ ( A4 = Ma3 ) @ ( heap_Time_return_o @ $true )
                                  @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ A4 @ Mi3 ) @ ( heap_Time_return_o @ $false )
                                    @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ A4 ) @ ( heap_Time_return_o @ $false )
                                      @ ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ A4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [H: nat] :
                                            ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ A4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [L2: nat] :
                                                ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList3 )
                                                @ ^ [Len: nat] :
                                                    ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                    @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList3 @ H )
                                                      @ ^ [Th: vEBT_VEBTi] : ( F2 @ Th @ L2 ) )
                                                    @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B4: $o,C3: $o] :
                        ( heap_Time_return_o
                        @ ( ( ( A4 = zero_zero_nat )
                           => B4 )
                          & ( ( A4 != zero_zero_nat )
                           => ( ( ( A4 = one_one_nat )
                               => C3 )
                              & ( A4 = one_one_nat ) ) ) ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_memberi ) ) ) ) ).

% vebt_memberi.fixp_induct
thf(fact_6920_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_6921_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_6922_tap__def,axiom,
    ( heap_T861569056895978319_VEBTi
    = ( ^ [F4: heap_e7401611519738050253t_unit > vEBT_VEBTi] :
          ( heap_T1489671670754571048_VEBTi
          @ ^ [H: heap_e7401611519738050253t_unit] : ( some_P696572436114257607it_nat @ ( produc8770982656060504474it_nat @ ( F4 @ H ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ) ).

% tap_def
thf(fact_6923_tap__def,axiom,
    ( heap_T3993282329578657722on_nat
    = ( ^ [F4: heap_e7401611519738050253t_unit > option_nat] :
          ( heap_T5286843759275942675on_nat
          @ ^ [H: heap_e7401611519738050253t_unit] : ( some_P2407035485129114418it_nat @ ( produc61566615109097733it_nat @ ( F4 @ H ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ) ).

% tap_def
thf(fact_6924_tap__def,axiom,
    ( heap_Time_tap_o
    = ( ^ [F4: heap_e7401611519738050253t_unit > $o] :
          ( heap_Time_Heap_o2
          @ ^ [H: heap_e7401611519738050253t_unit] : ( some_P468703482102919278it_nat @ ( produc6655106138504972685it_nat @ ( F4 @ H ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ) ).

% tap_def
thf(fact_6925_bezw__0,axiom,
    ! [X2: nat] :
      ( ( bezw @ X2 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_6926_effect__tapI,axiom,
    ! [H4: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,F: heap_e7401611519738050253t_unit > vEBT_VEBTi] :
      ( ( H4 = H2 )
     => ( ( R3
          = ( F @ H2 ) )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T861569056895978319_VEBTi @ F ) @ H2 @ H4 @ R3 @ one_one_nat ) ) ) ).

% effect_tapI
thf(fact_6927_effect__tapI,axiom,
    ! [H4: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R3: option_nat,F: heap_e7401611519738050253t_unit > option_nat] :
      ( ( H4 = H2 )
     => ( ( R3
          = ( F @ H2 ) )
       => ( heap_T306965388786959644on_nat @ ( heap_T3993282329578657722on_nat @ F ) @ H2 @ H4 @ R3 @ one_one_nat ) ) ) ).

% effect_tapI
thf(fact_6928_effect__tapI,axiom,
    ! [H4: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R3: $o,F: heap_e7401611519738050253t_unit > $o] :
      ( ( H4 = H2 )
     => ( ( R3
          = ( F @ H2 ) )
       => ( heap_Time_effect_o @ ( heap_Time_tap_o @ F ) @ H2 @ H4 @ R3 @ one_one_nat ) ) ) ).

% effect_tapI
thf(fact_6929_effect__tapE,axiom,
    ! [F: heap_e7401611519738050253t_unit > vEBT_VEBTi,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T861569056895978319_VEBTi @ F ) @ H2 @ H4 @ R3 @ N2 )
     => ~ ( ( H4 = H2 )
         => ( ( R3
              = ( F @ H2 ) )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_tapE
thf(fact_6930_effect__tapE,axiom,
    ! [F: heap_e7401611519738050253t_unit > option_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T3993282329578657722on_nat @ F ) @ H2 @ H4 @ R3 @ N2 )
     => ~ ( ( H4 = H2 )
         => ( ( R3
              = ( F @ H2 ) )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_tapE
thf(fact_6931_effect__tapE,axiom,
    ! [F: heap_e7401611519738050253t_unit > $o,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R3: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_Time_tap_o @ F ) @ H2 @ H4 @ R3 @ N2 )
     => ~ ( ( H4 = H2 )
         => ( ( R3
              = ( F @ H2 ) )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_tapE
thf(fact_6932_heap_Omono__body__fixp,axiom,
    ! [F5: ( 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] : ( F5 @ F4 @ X3 ) )
     => ( ( comple6805837186910174120on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ F5 )
        = ( F5 @ ( comple6805837186910174120on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ F5 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_6933_heap_Omono__body__fixp,axiom,
    ! [F5: ( 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] : ( F5 @ F4 @ X3 ) )
     => ( ( comple2405882057716616508Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ F5 )
        = ( F5 @ ( comple2405882057716616508Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ F5 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_6934_heap_Omono__body__fixp,axiom,
    ! [F5: ( 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] : ( F5 @ F4 @ X3 ) )
     => ( ( comple7072962176332223770_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi ) @ F5 )
        = ( F5 @ ( comple7072962176332223770_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi ) @ F5 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_6935_heap_Omono__body__fixp,axiom,
    ! [F5: ( 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] : ( F5 @ F4 @ X3 ) )
     => ( ( comple8068445680736955397on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat ) @ F5 )
        = ( F5 @ ( comple8068445680736955397on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat ) @ F5 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_6936_heap_Omono__body__fixp,axiom,
    ! [F5: ( 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] : ( F5 @ F4 @ X3 ) )
     => ( ( comple3202505432650402847Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o ) @ F5 )
        = ( F5 @ ( comple3202505432650402847Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o ) @ F5 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_6937_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_6938_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_6939_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_6940_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_6941_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_6942_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_6943_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_6944_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_6945_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_6946_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_6947_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr6911489093701683181Ti_nat @ ( produc2282297823089607884Ti_nat @ Xs2 @ Ys ) @ N2 )
        = ( produc7192665754729510430Ti_nat @ ( nth_VEBT_VEBTi @ 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_6948_pred__subset__eq2,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( ord_le4340812435750786203eger_o
        @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 )
        @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) )
      = ( ord_le653643898420964396nteger @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6949_pred__subset__eq2,axiom,
    ! [R2: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( ord_le8000401564054156549_nat_o
        @ ^ [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ R2 )
        @ ^ [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ S2 ) )
      = ( ord_le5966269811547037844et_nat @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6950_pred__subset__eq2,axiom,
    ! [R2: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( ord_le6753239538765779593_nat_o
        @ ^ [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ R2 )
        @ ^ [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ S2 ) )
      = ( ord_le4763372923235995152et_nat @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6951_pred__subset__eq2,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 )
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) )
      = ( ord_le8725513860283290265nt_int @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6952_pred__subset__eq2,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 )
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) )
      = ( ord_le135402666524580259nt_int @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6953_even__word__def,axiom,
    ( even_w9054469088133485505l_num1
    = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% even_word_def
thf(fact_6954_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_6955_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_6956_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_6957_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_6958_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_6959_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_6960_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_6961_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_6962_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_6963_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_6964_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_6965_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_6966_exhaustive__int_H_Ocases,axiom,
    ! [X2: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,D3: int,I3: int] :
          ( X2
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).

% exhaustive_int'.cases
thf(fact_6967_full__exhaustive__int_H_Ocases,axiom,
    ! [X2: produc2285326912895808259nt_int] :
      ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I3: int] :
          ( X2
         != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_6968_exhaustive__integer_H_Ocases,axiom,
    ! [X2: produc8763457246119570046nteger] :
      ~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
          ( X2
         != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_6969_full__exhaustive__integer_H_Ocases,axiom,
    ! [X2: produc1908205239877642774nteger] :
      ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
          ( X2
         != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_6970_pred__equals__eq2,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_6971_pred__equals__eq2,axiom,
    ! [R2: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( ( ^ [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_6972_pred__equals__eq2,axiom,
    ! [R2: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( ( ^ [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_6973_pred__equals__eq2,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_6974_pred__equals__eq2,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_6975_subrelI,axiom,
    ! [R3: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
      ( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
          ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R3 )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le653643898420964396nteger @ R3 @ S3 ) ) ).

% subrelI
thf(fact_6976_subrelI,axiom,
    ! [R3: set_Pr3286484037609594932et_nat,S3: set_Pr3286484037609594932et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
          ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R3 )
         => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le5966269811547037844et_nat @ R3 @ S3 ) ) ).

% subrelI
thf(fact_6977_subrelI,axiom,
    ! [R3: set_Pr8536935166611901872et_nat,S3: set_Pr8536935166611901872et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
          ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R3 )
         => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le4763372923235995152et_nat @ R3 @ S3 ) ) ).

% subrelI
thf(fact_6978_subrelI,axiom,
    ! [R3: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
      ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R3 )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le8725513860283290265nt_int @ R3 @ S3 ) ) ).

% subrelI
thf(fact_6979_subrelI,axiom,
    ! [R3: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
      ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R3 )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le135402666524580259nt_int @ R3 @ S3 ) ) ).

% subrelI
thf(fact_6980_top__empty__eq2,axiom,
    ( top_to8112647782992986859eger_o
    = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ top_to7512759353274530428nteger ) ) ) ).

% top_empty_eq2
thf(fact_6981_top__empty__eq2,axiom,
    ( top_to2428096842796733269_nat_o
    = ( ^ [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ top_to8753217654552796900et_nat ) ) ) ).

% top_empty_eq2
thf(fact_6982_top__empty__eq2,axiom,
    ( top_to7190503160269336793_nat_o
    = ( ^ [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ top_to1599102959340997728et_nat ) ) ) ).

% top_empty_eq2
thf(fact_6983_top__empty__eq2,axiom,
    ( top_to6261655714344447806_int_o
    = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ top_to3181862456159035625nt_int ) ) ) ).

% top_empty_eq2
thf(fact_6984_top__empty__eq2,axiom,
    ( top_to6513043852502318900_int_o
    = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ top_to2069137843433766899nt_int ) ) ) ).

% top_empty_eq2
thf(fact_6985_pred__subset__eq,axiom,
    ! [R2: set_nat,S2: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R2 )
        @ ^ [X: nat] : ( member_nat @ X @ S2 ) )
      = ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_6986_pred__subset__eq,axiom,
    ! [R2: set_VEBT_VEBT,S2: set_VEBT_VEBT] :
      ( ( ord_le418104280809901481VEBT_o
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ R2 )
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ S2 ) )
      = ( ord_le4337996190870823476T_VEBT @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_6987_pred__subset__eq,axiom,
    ! [R2: set_real,S2: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X: real] : ( member_real @ X @ R2 )
        @ ^ [X: real] : ( member_real @ X @ S2 ) )
      = ( ord_less_eq_set_real @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_6988_pred__subset__eq,axiom,
    ! [R2: set_int,S2: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X: int] : ( member_int @ X @ R2 )
        @ ^ [X: int] : ( member_int @ X @ S2 ) )
      = ( ord_less_eq_set_int @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_6989_exE__realizer,axiom,
    ! [P: nat > nat > $o,P4: product_prod_nat_nat,Q: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat] :
      ( ( P @ ( product_snd_nat_nat @ P4 ) @ ( product_fst_nat_nat @ P4 ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc2626176000494625587at_nat @ F @ P4 ) ) ) ) ).

% exE_realizer
thf(fact_6990_exE__realizer,axiom,
    ! [P: nat > nat > $o,P4: product_prod_nat_nat,Q: $o > $o,F: nat > nat > $o] :
      ( ( P @ ( product_snd_nat_nat @ P4 ) @ ( product_fst_nat_nat @ P4 ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc6081775807080527818_nat_o @ F @ P4 ) ) ) ) ).

% exE_realizer
thf(fact_6991_exE__realizer,axiom,
    ! [P: int > int > $o,P4: product_prod_int_int,Q: product_prod_int_int > $o,F: int > int > product_prod_int_int] :
      ( ( P @ ( product_snd_int_int @ P4 ) @ ( product_fst_int_int @ P4 ) )
     => ( ! [X3: int,Y3: int] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc4245557441103728435nt_int @ F @ P4 ) ) ) ) ).

% exE_realizer
thf(fact_6992_exE__realizer,axiom,
    ! [P: int > int > $o,P4: product_prod_int_int,Q: $o > $o,F: int > int > $o] :
      ( ( P @ ( product_snd_int_int @ P4 ) @ ( product_fst_int_int @ P4 ) )
     => ( ! [X3: int,Y3: int] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc4947309494688390418_int_o @ F @ P4 ) ) ) ) ).

% exE_realizer
thf(fact_6993_exE__realizer,axiom,
    ! [P: int > int > $o,P4: product_prod_int_int,Q: int > $o,F: int > int > int] :
      ( ( P @ ( product_snd_int_int @ P4 ) @ ( product_fst_int_int @ P4 ) )
     => ( ! [X3: int,Y3: int] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc8211389475949308722nt_int @ F @ P4 ) ) ) ) ).

% exE_realizer
thf(fact_6994_vebt__assn__raw_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y2: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X2 @ Xa )
        = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xa
                  = ( vEBT_Leafi @ Ai @ Bi ) )
               => ( Y2
                 != ( pure_assn
                    @ ( ( Ai = A3 )
                      & ( Bi = B2 ) ) ) ) ) )
       => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
             => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                  ( ( Xa
                    = ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
                 => ( Y2
                   != ( times_times_assn
                      @ ( times_times_assn
                        @ ( pure_assn
                          @ ( ( Mmoi2 = Mmo2 )
                            & ( Degi2 = Deg2 ) ) )
                        @ ( vEBT_vebt_assn_raw @ Summary2 @ 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 ) ) ) ) ) ) )
         => ( ( ? [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_6995_in__set__enumerate__eq,axiom,
    ! [P4: produc8025551001238799321T_VEBT,N2: nat,Xs2: list_VEBT_VEBT] :
      ( ( member8549952807677709168T_VEBT @ P4 @ ( set_Pr5984661752051438084T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( produc8575180428842422559T_VEBT @ P4 ) )
        & ( ord_less_nat @ ( produc8575180428842422559T_VEBT @ P4 ) @ ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) )
        & ( ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ ( produc8575180428842422559T_VEBT @ P4 ) @ N2 ) )
          = ( produc8172668247895388509T_VEBT @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_6996_in__set__enumerate__eq,axiom,
    ! [P4: produc214224863196444774_VEBTi,N2: nat,Xs2: list_VEBT_VEBTi] :
      ( ( member763447850065367567_VEBTi @ P4 @ ( set_Pr4207466110102731387_VEBTi @ ( enumerate_VEBT_VEBTi @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( produc8252055991070844170_VEBTi @ P4 ) )
        & ( ord_less_nat @ ( produc8252055991070844170_VEBTi @ P4 ) @ ( plus_plus_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ N2 ) )
        & ( ( nth_VEBT_VEBTi @ Xs2 @ ( minus_minus_nat @ ( produc8252055991070844170_VEBTi @ P4 ) @ N2 ) )
          = ( produc271786961351840588_VEBTi @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_6997_in__set__enumerate__eq,axiom,
    ! [P4: produc7716430852924023517t_real,N2: nat,Xs2: list_real] :
      ( ( member557208447399453958t_real @ P4 @ ( set_Pr7149346036329476978t_real @ ( enumerate_real @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_real @ P4 ) )
        & ( ord_less_nat @ ( product_fst_nat_real @ P4 ) @ ( plus_plus_nat @ ( size_size_list_real @ Xs2 ) @ N2 ) )
        & ( ( nth_real @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_real @ P4 ) @ N2 ) )
          = ( product_snd_nat_real @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_6998_in__set__enumerate__eq,axiom,
    ! [P4: product_prod_nat_o,N2: nat,Xs2: list_o] :
      ( ( member6310962623043647828_nat_o @ P4 @ ( set_Pr1291962091234853352_nat_o @ ( enumerate_o @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_o @ P4 ) )
        & ( ord_less_nat @ ( product_fst_nat_o @ P4 ) @ ( plus_plus_nat @ ( size_size_list_o @ Xs2 ) @ N2 ) )
        & ( ( nth_o @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_o @ P4 ) @ N2 ) )
          = ( product_snd_nat_o @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_6999_in__set__enumerate__eq,axiom,
    ! [P4: product_prod_nat_int,N2: nat,Xs2: list_int] :
      ( ( member4262671552274231302at_int @ P4 @ ( set_Pr1470767568048878706at_int @ ( enumerate_int @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_int @ P4 ) )
        & ( ord_less_nat @ ( product_fst_nat_int @ P4 ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) )
        & ( ( nth_int @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_int @ P4 ) @ N2 ) )
          = ( product_snd_nat_int @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7000_in__set__enumerate__eq,axiom,
    ! [P4: product_prod_nat_nat,N2: nat,Xs2: list_nat] :
      ( ( member8440522571783428010at_nat @ P4 @ ( set_Pr5648618587558075414at_nat @ ( enumerate_nat @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_nat @ P4 ) )
        & ( ord_less_nat @ ( product_fst_nat_nat @ P4 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) )
        & ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_nat @ P4 ) @ N2 ) )
          = ( product_snd_nat_nat @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7001_obtain__set__pred,axiom,
    ! [Z: nat,X2: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A2 @ X2 @ X_12 ) ) ) ) ).

% obtain_set_pred
thf(fact_7002_obtain__set__succ,axiom,
    ! [X2: nat,Z: nat,A2: set_nat,B5: set_nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ B5 )
         => ( ( A2 = B5 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A2 @ X2 @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_7003_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_7004_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ Xs2 )
              & ( ord_less_nat @ A @ X6 ) ) ) ) ).

% succ_none_empty
thf(fact_7005_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ Xs2 )
              & ( ord_less_nat @ X6 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_7006_finite__option__UNIV,axiom,
    ( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
    = ( finite_finite_int @ top_top_set_int ) ) ).

% finite_option_UNIV
thf(fact_7007_finite__option__UNIV,axiom,
    ( ( finite2569390945932476949omplex @ top_to6180147692022559204omplex )
    = ( finite3207457112153483333omplex @ top_top_set_complex ) ) ).

% finite_option_UNIV
thf(fact_7008_finite__option__UNIV,axiom,
    ( ( finite6785661671136154180nteger @ top_to5929521628599800467nteger )
    = ( finite6017078050557962740nteger @ top_to4645266643341252675nteger ) ) ).

% finite_option_UNIV
thf(fact_7009_finite__option__UNIV,axiom,
    ( ( finite5731380839103853331n_real @ top_to853713521313446370n_real )
    = ( finite_finite_real @ top_top_set_real ) ) ).

% finite_option_UNIV
thf(fact_7010_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_7011_finite__option__UNIV,axiom,
    ( ( finite2053776701332276152n_char @ top_to576892095687177735n_char )
    = ( finite_finite_char @ top_top_set_char ) ) ).

% finite_option_UNIV
thf(fact_7012_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7013_List_Ofinite__set,axiom,
    ! [Xs2: list_real] : ( finite_finite_real @ ( set_real2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7014_List_Ofinite__set,axiom,
    ! [Xs2: list_P8527749157015355191n_assn] : ( finite5137929494490007386n_assn @ ( set_Pr1139785259514867910n_assn @ Xs2 ) ) ).

% List.finite_set
thf(fact_7015_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7016_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7017_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7018_List_Ofinite__set,axiom,
    ! [Xs2: list_Code_integer] : ( finite6017078050557962740nteger @ ( set_Code_integer2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7019_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_7020_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_7021_false__rule,axiom,
    ! [C: heap_T2636463487746394924on_nat,Q: option_nat > assn] : ( hoare_7629718768684598413on_nat @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_7022_false__rule,axiom,
    ! [C: heap_Time_Heap_nat,Q: nat > assn] : ( hoare_3067605981109127869le_nat @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_7023_false__rule,axiom,
    ! [C: heap_Time_Heap_o,Q: $o > assn] : ( hoare_hoare_triple_o @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_7024_false__rule,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] : ( hoare_1429296392585015714_VEBTi @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_7025_false__rule,axiom,
    ! [C: heap_T5738788834812785303t_unit,Q: product_unit > assn] : ( hoare_8945653483474564448t_unit @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_7026_length__enumerate,axiom,
    ! [N2: nat,Xs2: list_real] :
      ( ( size_s7910714270633306959t_real @ ( enumerate_real @ N2 @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_enumerate
thf(fact_7027_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_7028_length__enumerate,axiom,
    ! [N2: nat,Xs2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_enumerate
thf(fact_7029_length__enumerate,axiom,
    ! [N2: nat,Xs2: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N2 @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_enumerate
thf(fact_7030_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_7031_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_7032_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_7033_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_7034_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_7035_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_7036_snga__same__false,axiom,
    ! [P4: array_VEBT_VEBTi,X2: list_VEBT_VEBTi,Y2: list_VEBT_VEBTi] :
      ( ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ P4 @ X2 ) @ ( snga_assn_VEBT_VEBTi @ P4 @ Y2 ) )
      = bot_bot_assn ) ).

% snga_same_false
thf(fact_7037_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_int,S5: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S4: set_int] :
              ( ( minus_minus_set_int @ S4 @ S2 )
              = ( minus_minus_set_int @ S5 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7038_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_complex,S5: set_complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [S4: set_complex] :
              ( ( minus_811609699411566653omplex @ S4 @ S2 )
              = ( minus_811609699411566653omplex @ S5 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7039_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_Code_integer,S5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S4: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S4 @ S2 )
              = ( minus_2355218937544613996nteger @ S5 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7040_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_nat,S5: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S4: set_nat] :
              ( ( minus_minus_set_nat @ S4 @ S2 )
              = ( minus_minus_set_nat @ S5 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7041_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N9: set_nat] :
        ? [M7: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N9 )
         => ( ord_less_nat @ X @ M7 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_7042_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_7043_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N9: set_nat] :
        ? [M7: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N9 )
         => ( ord_less_eq_nat @ X @ M7 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_7044_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7045_finite__list,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ? [Xs3: list_real] :
          ( ( set_real2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7046_finite__list,axiom,
    ! [A2: set_Pr5949110396991348497n_assn] :
      ( ( finite5137929494490007386n_assn @ A2 )
     => ? [Xs3: list_P8527749157015355191n_assn] :
          ( ( set_Pr1139785259514867910n_assn @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7047_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7048_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7049_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7050_finite__list,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ? [Xs3: list_Code_integer] :
          ( ( set_Code_integer2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7051_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K4: nat] :
            ( ( P @ K4 )
            & ( ord_less_nat @ K4 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_7052_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_7053_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7054_finite__lists__length__eq,axiom,
    ! [A2: set_Pr5949110396991348497n_assn,N2: nat] :
      ( ( finite5137929494490007386n_assn @ A2 )
     => ( finite1351478129840809056n_assn
        @ ( collec8177951099088521122n_assn
          @ ^ [Xs: list_P8527749157015355191n_assn] :
              ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ A2 )
              & ( ( size_s6829681357464350627n_assn @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7055_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7056_finite__lists__length__eq,axiom,
    ! [A2: set_Code_integer,N2: nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A2 )
              & ( ( size_s3445333598471063425nteger @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7057_finite__lists__length__eq,axiom,
    ! [A2: set_real,N2: nat] :
      ( ( finite_finite_real @ A2 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
              & ( ( size_size_list_real @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7058_finite__lists__length__eq,axiom,
    ! [A2: set_o,N2: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ( size_size_list_o @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7059_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ( size_size_list_nat @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7060_finite__lists__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ( size_size_list_int @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7061_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_7062_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_7063_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7064_finite__lists__length__le,axiom,
    ! [A2: set_Pr5949110396991348497n_assn,N2: nat] :
      ( ( finite5137929494490007386n_assn @ A2 )
     => ( finite1351478129840809056n_assn
        @ ( collec8177951099088521122n_assn
          @ ^ [Xs: list_P8527749157015355191n_assn] :
              ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6829681357464350627n_assn @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7065_finite__lists__length__le,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7066_finite__lists__length__le,axiom,
    ! [A2: set_Code_integer,N2: nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3445333598471063425nteger @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7067_finite__lists__length__le,axiom,
    ! [A2: set_real,N2: nat] :
      ( ( finite_finite_real @ A2 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7068_finite__lists__length__le,axiom,
    ! [A2: set_o,N2: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7069_finite__lists__length__le,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7070_finite__lists__length__le,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7071_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Vd: $o,Ve: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va @ Vb @ Vc2 ) @ ( vEBT_Leafi @ Vd @ Ve ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(3)
thf(fact_7072_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option4927543243414619207at_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V @ Va @ Vb @ Vc2 ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(4)
thf(fact_7073_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_7074_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_7075_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N2 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_7076_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z2: real] :
              ( ( power_power_real @ Z2 @ N2 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_7077_exE__realizer_H,axiom,
    ! [P: uint32 > uint32 > $o,P4: produc827990862158126777uint32] :
      ( ( P @ ( produc1510406741064981791uint32 @ P4 ) @ ( produc9004433772639906525uint32 @ P4 ) )
     => ~ ! [X3: uint32,Y3: uint32] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7078_exE__realizer_H,axiom,
    ! [P: nat > nat > $o,P4: product_prod_nat_nat] :
      ( ( P @ ( product_snd_nat_nat @ P4 ) @ ( product_fst_nat_nat @ P4 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7079_exE__realizer_H,axiom,
    ! [P: set_nat > heap_e7401611519738050253t_unit > $o,P4: produc3658429121746597890et_nat] :
      ( ( P @ ( produc8586169260539613262et_nat @ P4 ) @ ( produc1824681642469235216et_nat @ P4 ) )
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y3: set_nat] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7080_exE__realizer_H,axiom,
    ! [P: int > int > $o,P4: product_prod_int_int] :
      ( ( P @ ( product_snd_int_int @ P4 ) @ ( product_fst_int_int @ P4 ) )
     => ~ ! [X3: int,Y3: int] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7081_exE__realizer_H,axiom,
    ! [P: code_integer > code_integer > $o,P4: produc8923325533196201883nteger] :
      ( ( P @ ( produc6174133586879617921nteger @ P4 ) @ ( produc8508995932063986495nteger @ P4 ) )
     => ~ ! [X3: code_integer,Y3: code_integer] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7082_exE__realizer_H,axiom,
    ! [P: assn > assn > $o,P4: produc6575502325842934193n_assn] :
      ( ( P @ ( produc2051961928117032727n_assn @ P4 ) @ ( produc9167289414957590229n_assn @ P4 ) )
     => ~ ! [X3: assn,Y3: assn] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7083_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_7084_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_7085_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_7086_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_7087_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_7088_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_7089_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X2: option_num,R2: option_num > $o,F: option_num,G: num > option_num] :
      ( ( case_option_o_num @ P @ Q @ X2 )
     => ( ( P
         => ( R2 @ F ) )
       => ( ! [Q3: num] :
              ( ( Q @ Q3 )
             => ( R2 @ ( G @ Q3 ) ) )
         => ( R2 @ ( case_o6005452278849405969um_num @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7090_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X2: option_num,R2: num > $o,F: num,G: num > num] :
      ( ( case_option_o_num @ P @ Q @ X2 )
     => ( ( P
         => ( R2 @ F ) )
       => ( ! [Q3: num] :
              ( ( Q @ Q3 )
             => ( R2 @ ( G @ Q3 ) ) )
         => ( R2 @ ( case_option_num_num @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7091_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X2: option_num,R2: int > $o,F: int,G: num > int] :
      ( ( case_option_o_num @ P @ Q @ X2 )
     => ( ( P
         => ( R2 @ F ) )
       => ( ! [Q3: num] :
              ( ( Q @ Q3 )
             => ( R2 @ ( G @ Q3 ) ) )
         => ( R2 @ ( case_option_int_num @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7092_disjE__realizer2,axiom,
    ! [P: $o,Q: product_prod_nat_nat > $o,X2: option4927543243414619207at_nat,R2: 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
         => ( R2 @ F ) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( ( Q @ Q3 )
             => ( R2 @ ( G @ Q3 ) ) )
         => ( R2 @ ( case_o1442776274061689234at_nat @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7093_disjE__realizer2,axiom,
    ! [P: $o,Q: product_prod_nat_nat > $o,X2: option4927543243414619207at_nat,R2: $o > $o,F: $o,G: product_prod_nat_nat > $o] :
      ( ( case_o184042715313410164at_nat @ P @ Q @ X2 )
     => ( ( P
         => ( R2 @ F ) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( ( Q @ Q3 )
             => ( R2 @ ( G @ Q3 ) ) )
         => ( R2 @ ( case_o184042715313410164at_nat @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7094_conjI__realizer,axiom,
    ! [P: uint32 > $o,P4: uint32,Q: uint32 > $o,Q2: uint32] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc9004433772639906525uint32 @ ( produc1400373151660368625uint32 @ P4 @ Q2 ) ) )
          & ( Q @ ( produc1510406741064981791uint32 @ ( produc1400373151660368625uint32 @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7095_conjI__realizer,axiom,
    ! [P: nat > $o,P4: nat,Q: nat > $o,Q2: nat] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P4 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7096_conjI__realizer,axiom,
    ! [P: int > $o,P4: int,Q: int > $o,Q2: int] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P4 @ Q2 ) ) )
          & ( Q @ ( product_snd_int_int @ ( product_Pair_int_int @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7097_conjI__realizer,axiom,
    ! [P: code_integer > $o,P4: code_integer,Q: code_integer > $o,Q2: code_integer] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc8508995932063986495nteger @ ( produc1086072967326762835nteger @ P4 @ Q2 ) ) )
          & ( Q @ ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7098_conjI__realizer,axiom,
    ! [P: assn > $o,P4: assn,Q: assn > $o,Q2: assn] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc9167289414957590229n_assn @ ( produc118845697133431529n_assn @ P4 @ Q2 ) ) )
          & ( Q @ ( produc2051961928117032727n_assn @ ( produc118845697133431529n_assn @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7099_conjI__realizer,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,P4: heap_e7401611519738050253t_unit,Q: set_nat > $o,Q2: set_nat] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ P4 @ Q2 ) ) )
          & ( Q @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7100_conjI__realizer,axiom,
    ! [P: ( int > option6357759511663192854e_term ) > $o,P4: int > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q2: product_prod_int_int] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ P4 @ Q2 ) ) )
          & ( Q @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7101_conjI__realizer,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P4: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ P4 @ Q2 ) ) )
          & ( Q @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7102_conjI__realizer,axiom,
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,P4: produc6241069584506657477e_term > option6357759511663192854e_term,Q: produc8923325533196201883nteger > $o,Q2: produc8923325533196201883nteger] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ P4 @ Q2 ) ) )
          & ( Q @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7103_conjI__realizer,axiom,
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,P4: produc8551481072490612790e_term > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q2: product_prod_int_int] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ P4 @ Q2 ) ) )
          & ( Q @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7104_exI__realizer,axiom,
    ! [P: uint32 > uint32 > $o,Y2: uint32,X2: uint32] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc1510406741064981791uint32 @ ( produc1400373151660368625uint32 @ X2 @ Y2 ) ) @ ( produc9004433772639906525uint32 @ ( produc1400373151660368625uint32 @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7105_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_7106_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_7107_exI__realizer,axiom,
    ! [P: code_integer > code_integer > $o,Y2: code_integer,X2: code_integer] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ X2 @ Y2 ) ) @ ( produc8508995932063986495nteger @ ( produc1086072967326762835nteger @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7108_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_7109_exI__realizer,axiom,
    ! [P: set_nat > heap_e7401611519738050253t_unit > $o,Y2: set_nat,X2: heap_e7401611519738050253t_unit] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y2 ) ) @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7110_exI__realizer,axiom,
    ! [P: product_prod_int_int > ( int > option6357759511663192854e_term ) > $o,Y2: product_prod_int_int,X2: int > option6357759511663192854e_term] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) ) @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7111_exI__realizer,axiom,
    ! [P: produc3658429121746597890et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y2: produc3658429121746597890et_nat,X2: produc3658429121746597890et_nat > $o] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) ) @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7112_exI__realizer,axiom,
    ! [P: produc8923325533196201883nteger > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,Y2: produc8923325533196201883nteger,X2: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) ) @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7113_exI__realizer,axiom,
    ! [P: product_prod_int_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Y2: product_prod_int_int,X2: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) ) @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7114_finite__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B5 ) ) )
      = ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_7115_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B5: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B5 ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_7116_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B5: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B5 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_7117_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B5 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_7118_finite__Diff__insert,axiom,
    ! [A2: set_Code_integer,A: code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ B5 ) ) )
      = ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_7119_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B5: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_7120_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_7121_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_7122_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7123_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7124_finite__Collect__subsets,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B6: set_Code_integer] : ( ord_le7084787975880047091nteger @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7125_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7126_finite__Collect__not,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ~ ( P @ X ) ) )
        = ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int ) ) ) ).

% finite_Collect_not
thf(fact_7127_finite__Collect__not,axiom,
    ! [P: int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
                ~ ( P @ X ) ) )
        = ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_Collect_not
thf(fact_7128_finite__Collect__not,axiom,
    ! [P: complex > $o] :
      ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X: complex] :
                ~ ( P @ X ) ) )
        = ( finite3207457112153483333omplex @ top_top_set_complex ) ) ) ).

% finite_Collect_not
thf(fact_7129_finite__Collect__not,axiom,
    ! [P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ~ ( P @ X ) ) )
        = ( finite6017078050557962740nteger @ top_to4645266643341252675nteger ) ) ) ).

% finite_Collect_not
thf(fact_7130_finite__Collect__not,axiom,
    ! [P: real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
                ~ ( P @ X ) ) )
        = ( finite_finite_real @ top_top_set_real ) ) ) ).

% finite_Collect_not
thf(fact_7131_finite__Collect__not,axiom,
    ! [P: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_7132_finite__Collect__not,axiom,
    ! [P: char > $o] :
      ( ( finite_finite_char @ ( collect_char @ P ) )
     => ( ( finite_finite_char
          @ ( collect_char
            @ ^ [X: char] :
                ~ ( P @ X ) ) )
        = ( finite_finite_char @ top_top_set_char ) ) ) ).

% finite_Collect_not
thf(fact_7133_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_7134_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_7135_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_7136_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_7137_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_7138_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_7139_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_7140_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_7141_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_7142_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_7143_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_7144_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_7145_finite__Diff2,axiom,
    ! [B5: set_int,A2: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) )
        = ( finite_finite_int @ A2 ) ) ) ).

% finite_Diff2
thf(fact_7146_finite__Diff2,axiom,
    ! [B5: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
        = ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_Diff2
thf(fact_7147_finite__Diff2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
        = ( finite6017078050557962740nteger @ A2 ) ) ) ).

% finite_Diff2
thf(fact_7148_finite__Diff2,axiom,
    ! [B5: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_7149_finite__Diff,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_7150_finite__Diff,axiom,
    ! [A2: set_complex,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_7151_finite__Diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_7152_finite__Diff,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_7153_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_7154_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_7155_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( ( finite306553202115118035t_real @ M8 )
     => ? [N4: nat] :
        ! [X6: list_real] :
          ( ( member_list_real @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_real @ X6 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_7156_finite__maxlen,axiom,
    ! [M8: set_list_o] :
      ( ( finite_finite_list_o @ M8 )
     => ? [N4: nat] :
        ! [X6: list_o] :
          ( ( member_list_o @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_o @ X6 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_7157_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M8 )
     => ? [N4: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_7158_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( ( finite3922522038869484883st_int @ M8 )
     => ? [N4: nat] :
        ! [X6: list_int] :
          ( ( member_list_int @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_int @ X6 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_7159_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_7160_not__finite__existsD,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ? [X_12: product_prod_int_int] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_7161_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_12: nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_7162_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_12: int] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_7163_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_12: complex] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_7164_not__finite__existsD,axiom,
    ! [P: code_integer > $o] :
      ( ~ ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
     => ? [X_12: code_integer] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_7165_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_nat,R2: vEBT_VEBT > nat > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B5 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7166_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B5: set_nat,R2: real > nat > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B5 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7167_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_int,R2: vEBT_VEBT > int > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B5 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7168_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B5: set_int,R2: real > int > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B5 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7169_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_complex,R2: vEBT_VEBT > complex > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ? [Xa2: complex] :
                  ( ( member_complex @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B5 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7170_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B5: set_complex,R2: real > complex > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ? [Xa2: complex] :
                  ( ( member_complex @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B5 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7171_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_Code_integer,R2: vEBT_VEBT > code_integer > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B5 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7172_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B5: set_Code_integer,R2: real > code_integer > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B5 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7173_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B5: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B5 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7174_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B5: set_int,R2: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B5 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B5 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7175_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ( ord_less_eq_real @ A @ X3 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7176_finite__has__maximal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ A @ A2 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ( ord_le3102999989581377725nteger @ A @ X3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7177_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ( ord_less_eq_set_int @ A @ X3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7178_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7179_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7180_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7181_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7182_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ( ord_less_eq_real @ X3 @ A )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7183_finite__has__minimal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ A @ A2 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ( ord_le3102999989581377725nteger @ X3 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7184_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ( ord_less_eq_set_int @ X3 @ A )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7185_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7186_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7187_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7188_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7189_finite__subset,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( finite_finite_nat @ B5 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_7190_finite__subset,axiom,
    ! [A2: set_complex,B5: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B5 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_7191_finite__subset,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( finite6017078050557962740nteger @ A2 ) ) ) ).

% finite_subset
thf(fact_7192_finite__subset,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( finite_finite_int @ B5 )
       => ( finite_finite_int @ A2 ) ) ) ).

% finite_subset
thf(fact_7193_infinite__super,axiom,
    ! [S2: set_nat,T5: set_nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ T5 ) ) ) ).

% infinite_super
thf(fact_7194_infinite__super,axiom,
    ! [S2: set_complex,T5: set_complex] :
      ( ( ord_le211207098394363844omplex @ S2 @ T5 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ T5 ) ) ) ).

% infinite_super
thf(fact_7195_infinite__super,axiom,
    ! [S2: set_Code_integer,T5: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
     => ( ~ ( finite6017078050557962740nteger @ S2 )
       => ~ ( finite6017078050557962740nteger @ T5 ) ) ) ).

% infinite_super
thf(fact_7196_infinite__super,axiom,
    ! [S2: set_int,T5: set_int] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ~ ( finite_finite_int @ S2 )
       => ~ ( finite_finite_int @ T5 ) ) ) ).

% infinite_super
thf(fact_7197_rev__finite__subset,axiom,
    ! [B5: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_7198_rev__finite__subset,axiom,
    ! [B5: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_7199_rev__finite__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( finite6017078050557962740nteger @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_7200_rev__finite__subset,axiom,
    ! [B5: set_int,A2: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A2 @ B5 )
       => ( finite_finite_int @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_7201_Diff__infinite__finite,axiom,
    ! [T5: set_int,S2: set_int] :
      ( ( finite_finite_int @ T5 )
     => ( ~ ( finite_finite_int @ S2 )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T5 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_7202_Diff__infinite__finite,axiom,
    ! [T5: set_complex,S2: set_complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T5 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_7203_Diff__infinite__finite,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ~ ( finite6017078050557962740nteger @ S2 )
       => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S2 @ T5 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_7204_Diff__infinite__finite,axiom,
    ! [T5: set_nat,S2: set_nat] :
      ( ( finite_finite_nat @ T5 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T5 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_7205_finite__has__minimal,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7206_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7207_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7208_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7209_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7210_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7211_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7212_finite__has__maximal,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7213_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7214_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7215_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7216_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7217_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7218_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7219_finite__subset__induct_H,axiom,
    ! [F5: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( ord_le4337996190870823476T_VEBT @ F5 @ A2 )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( member_VEBT_VEBT @ A3 @ A2 )
                 => ( ( ord_le4337996190870823476T_VEBT @ F6 @ A2 )
                   => ( ~ ( member_VEBT_VEBT @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_VEBT_VEBT @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_7220_finite__subset__induct_H,axiom,
    ! [F5: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( ord_le211207098394363844omplex @ F5 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( member_complex @ A3 @ A2 )
                 => ( ( ord_le211207098394363844omplex @ F6 @ A2 )
                   => ( ~ ( member_complex @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_complex @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_7221_finite__subset__induct_H,axiom,
    ! [F5: set_Code_integer,A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger @ F5 @ A2 )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A3 @ A2 )
                 => ( ( ord_le7084787975880047091nteger @ F6 @ A2 )
                   => ( ~ ( member_Code_integer @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_Code_integer @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_7222_finite__subset__induct_H,axiom,
    ! [F5: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat @ F5 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A3 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F6 @ A2 )
                   => ( ~ ( member_nat @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_nat @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_7223_finite__subset__induct_H,axiom,
    ! [F5: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F5 )
     => ( ( ord_less_eq_set_real @ F5 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( member_real @ A3 @ A2 )
                 => ( ( ord_less_eq_set_real @ F6 @ A2 )
                   => ( ~ ( member_real @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_real @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_7224_finite__subset__induct_H,axiom,
    ! [F5: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( ord_less_eq_set_int @ F5 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A3 @ A2 )
                 => ( ( ord_less_eq_set_int @ F6 @ A2 )
                   => ( ~ ( member_int @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_int @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_7225_finite__subset__induct,axiom,
    ! [F5: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( ord_le4337996190870823476T_VEBT @ F5 @ A2 )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( member_VEBT_VEBT @ A3 @ A2 )
                 => ( ~ ( member_VEBT_VEBT @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_VEBT_VEBT @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_7226_finite__subset__induct,axiom,
    ! [F5: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( ord_le211207098394363844omplex @ F5 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( member_complex @ A3 @ A2 )
                 => ( ~ ( member_complex @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_complex @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_7227_finite__subset__induct,axiom,
    ! [F5: set_Code_integer,A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger @ F5 @ A2 )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A3 @ A2 )
                 => ( ~ ( member_Code_integer @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_Code_integer @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_7228_finite__subset__induct,axiom,
    ! [F5: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat @ F5 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A3 @ A2 )
                 => ( ~ ( member_nat @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_nat @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_7229_finite__subset__induct,axiom,
    ! [F5: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F5 )
     => ( ( ord_less_eq_set_real @ F5 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( member_real @ A3 @ A2 )
                 => ( ~ ( member_real @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_real @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_7230_finite__subset__induct,axiom,
    ! [F5: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( ord_less_eq_set_int @ F5 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A3 @ A2 )
                 => ( ~ ( member_int @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_int @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_7231_infinite__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ~ ( finite5795047828879050333T_VEBT @ S2 )
     => ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% infinite_remove
thf(fact_7232_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S2 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_7233_infinite__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S2 )
     => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ).

% infinite_remove
thf(fact_7234_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ ( finite_finite_int @ S2 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_7235_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ ( finite_finite_real @ S2 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_7236_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_7237_infinite__coinduct,axiom,
    ! [X8: set_VEBT_VEBT > $o,A2: set_VEBT_VEBT] :
      ( ( X8 @ A2 )
     => ( ! [A8: set_VEBT_VEBT] :
            ( ( X8 @ A8 )
           => ? [X6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X6 @ A8 )
                & ( ( X8 @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) )
                  | ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
       => ~ ( finite5795047828879050333T_VEBT @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_7238_infinite__coinduct,axiom,
    ! [X8: set_complex > $o,A2: set_complex] :
      ( ( X8 @ A2 )
     => ( ! [A8: set_complex] :
            ( ( X8 @ A8 )
           => ? [X6: complex] :
                ( ( member_complex @ X6 @ A8 )
                & ( ( X8 @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_7239_infinite__coinduct,axiom,
    ! [X8: set_Code_integer > $o,A2: set_Code_integer] :
      ( ( X8 @ A2 )
     => ( ! [A8: set_Code_integer] :
            ( ( X8 @ A8 )
           => ? [X6: code_integer] :
                ( ( member_Code_integer @ X6 @ A8 )
                & ( ( X8 @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) )
                  | ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
       => ~ ( finite6017078050557962740nteger @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_7240_infinite__coinduct,axiom,
    ! [X8: set_int > $o,A2: set_int] :
      ( ( X8 @ A2 )
     => ( ! [A8: set_int] :
            ( ( X8 @ A8 )
           => ? [X6: int] :
                ( ( member_int @ X6 @ A8 )
                & ( ( X8 @ ( minus_minus_set_int @ A8 @ ( insert_int @ X6 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A8 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_7241_infinite__coinduct,axiom,
    ! [X8: set_real > $o,A2: set_real] :
      ( ( X8 @ A2 )
     => ( ! [A8: set_real] :
            ( ( X8 @ A8 )
           => ? [X6: real] :
                ( ( member_real @ X6 @ A8 )
                & ( ( X8 @ ( minus_minus_set_real @ A8 @ ( insert_real @ X6 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A8 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_7242_infinite__coinduct,axiom,
    ! [X8: set_nat > $o,A2: set_nat] :
      ( ( X8 @ A2 )
     => ( ! [A8: set_nat] :
            ( ( X8 @ A8 )
           => ? [X6: nat] :
                ( ( member_nat @ X6 @ A8 )
                & ( ( X8 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_7243_finite__empty__induct,axiom,
    ! [A2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: vEBT_VEBT,A8: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A8 )
             => ( ( member_VEBT_VEBT @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
         => ( P @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% finite_empty_induct
thf(fact_7244_finite__empty__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: complex,A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( member_complex @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_7245_finite__empty__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ( member_Code_integer @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
         => ( P @ bot_bo3990330152332043303nteger ) ) ) ) ).

% finite_empty_induct
thf(fact_7246_finite__empty__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( member_int @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_7247_finite__empty__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( member_real @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_7248_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( member_nat @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_7249_remove__induct,axiom,
    ! [P: set_VEBT_VEBT > $o,B5: set_VEBT_VEBT] :
      ( ( P @ bot_bo8194388402131092736T_VEBT )
     => ( ( ~ ( finite5795047828879050333T_VEBT @ B5 )
         => ( P @ B5 ) )
       => ( ! [A8: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A8 )
             => ( ( A8 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A8 @ B5 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ A8 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_7250_remove__induct,axiom,
    ! [P: set_complex > $o,B5: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B5 )
         => ( P @ B5 ) )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B5 )
                 => ( ! [X6: complex] :
                        ( ( member_complex @ X6 @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_7251_remove__induct,axiom,
    ! [P: set_Code_integer > $o,B5: set_Code_integer] :
      ( ( P @ bot_bo3990330152332043303nteger )
     => ( ( ~ ( finite6017078050557962740nteger @ B5 )
         => ( P @ B5 ) )
       => ( ! [A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ( A8 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A8 @ B5 )
                 => ( ! [X6: code_integer] :
                        ( ( member_Code_integer @ X6 @ A8 )
                       => ( P @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_7252_remove__induct,axiom,
    ! [P: set_real > $o,B5: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B5 )
         => ( P @ B5 ) )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B5 )
                 => ( ! [X6: real] :
                        ( ( member_real @ X6 @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_7253_remove__induct,axiom,
    ! [P: set_nat > $o,B5: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B5 )
         => ( P @ B5 ) )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B5 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_7254_remove__induct,axiom,
    ! [P: set_int > $o,B5: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B5 )
         => ( P @ B5 ) )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B5 )
                 => ( ! [X6: int] :
                        ( ( member_int @ X6 @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_7255_finite__remove__induct,axiom,
    ! [B5: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [A8: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A8 )
             => ( ( A8 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A8 @ B5 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ A8 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_7256_finite__remove__induct,axiom,
    ! [B5: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B5 )
                 => ( ! [X6: complex] :
                        ( ( member_complex @ X6 @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_7257_finite__remove__induct,axiom,
    ! [B5: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ( A8 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A8 @ B5 )
                 => ( ! [X6: code_integer] :
                        ( ( member_Code_integer @ X6 @ A8 )
                       => ( P @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_7258_finite__remove__induct,axiom,
    ! [B5: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B5 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B5 )
                 => ( ! [X6: real] :
                        ( ( member_real @ X6 @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_7259_finite__remove__induct,axiom,
    ! [B5: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B5 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_7260_finite__remove__induct,axiom,
    ! [B5: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B5 )
                 => ( ! [X6: int] :
                        ( ( member_int @ X6 @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_7261_finite__induct__select,axiom,
    ! [S2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [T6: set_VEBT_VEBT] :
              ( ( ord_le3480810397992357184T_VEBT @ T6 @ S2 )
             => ( ( P @ T6 )
               => ? [X6: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X6 @ ( minus_5127226145743854075T_VEBT @ S2 @ T6 ) )
                    & ( P @ ( insert_VEBT_VEBT @ X6 @ T6 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_7262_finite__induct__select,axiom,
    ! [S2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T6: set_complex] :
              ( ( ord_less_set_complex @ T6 @ S2 )
             => ( ( P @ T6 )
               => ? [X6: complex] :
                    ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ S2 @ T6 ) )
                    & ( P @ ( insert_complex @ X6 @ T6 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_7263_finite__induct__select,axiom,
    ! [S2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [T6: set_Code_integer] :
              ( ( ord_le1307284697595431911nteger @ T6 @ S2 )
             => ( ( P @ T6 )
               => ? [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ ( minus_2355218937544613996nteger @ S2 @ T6 ) )
                    & ( P @ ( insert_Code_integer @ X6 @ T6 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_7264_finite__induct__select,axiom,
    ! [S2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T6: set_int] :
              ( ( ord_less_set_int @ T6 @ S2 )
             => ( ( P @ T6 )
               => ? [X6: int] :
                    ( ( member_int @ X6 @ ( minus_minus_set_int @ S2 @ T6 ) )
                    & ( P @ ( insert_int @ X6 @ T6 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_7265_finite__induct__select,axiom,
    ! [S2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T6: set_real] :
              ( ( ord_less_set_real @ T6 @ S2 )
             => ( ( P @ T6 )
               => ? [X6: real] :
                    ( ( member_real @ X6 @ ( minus_minus_set_real @ S2 @ T6 ) )
                    & ( P @ ( insert_real @ X6 @ T6 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_7266_finite__induct__select,axiom,
    ! [S2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T6: set_nat] :
              ( ( ord_less_set_nat @ T6 @ S2 )
             => ( ( P @ T6 )
               => ? [X6: nat] :
                    ( ( member_nat @ X6 @ ( minus_minus_set_nat @ S2 @ T6 ) )
                    & ( P @ ( insert_nat @ X6 @ T6 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_7267_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_7268_set__encode__insert,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N2 @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_7269_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_7270_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_7271_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_7272_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_7273_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_7274_add__mset__in__multiset,axiom,
    ! [M8: product_prod_int_int > nat,A: 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 = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7275_add__mset__in__multiset,axiom,
    ! [M8: nat > nat,A: 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 = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7276_add__mset__in__multiset,axiom,
    ! [M8: int > nat,A: 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 = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7277_add__mset__in__multiset,axiom,
    ! [M8: complex > nat,A: 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 = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7278_add__mset__in__multiset,axiom,
    ! [M8: code_integer > nat,A: 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 = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7279_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B2: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ A8 )
                   => ( ord_le6747313008572928689nteger @ X6 @ B2 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7280_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B2: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X6: real] :
                    ( ( member_real @ X6 @ A8 )
                   => ( ord_less_real @ X6 @ B2 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7281_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B2: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X6: rat] :
                    ( ( member_rat @ X6 @ A8 )
                   => ( ord_less_rat @ X6 @ B2 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7282_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B2: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X6: num] :
                    ( ( member_num @ X6 @ A8 )
                   => ( ord_less_num @ X6 @ B2 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7283_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B2: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A8 )
                   => ( ord_less_nat @ X6 @ B2 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7284_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B2: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X6: int] :
                    ( ( member_int @ X6 @ A8 )
                   => ( ord_less_int @ X6 @ B2 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7285_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_7286_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).

% even_set_encode_iff
thf(fact_7287_infinite__growing,axiom,
    ! [X8: set_Code_integer] :
      ( ( X8 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X8 )
           => ? [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ X8 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
       => ~ ( finite6017078050557962740nteger @ X8 ) ) ) ).

% infinite_growing
thf(fact_7288_infinite__growing,axiom,
    ! [X8: set_real] :
      ( ( X8 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X8 )
           => ? [Xa2: real] :
                ( ( member_real @ Xa2 @ X8 )
                & ( ord_less_real @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_real @ X8 ) ) ) ).

% infinite_growing
thf(fact_7289_infinite__growing,axiom,
    ! [X8: set_rat] :
      ( ( X8 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X8 )
           => ? [Xa2: rat] :
                ( ( member_rat @ Xa2 @ X8 )
                & ( ord_less_rat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_rat @ X8 ) ) ) ).

% infinite_growing
thf(fact_7290_infinite__growing,axiom,
    ! [X8: set_num] :
      ( ( X8 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X8 )
           => ? [Xa2: num] :
                ( ( member_num @ Xa2 @ X8 )
                & ( ord_less_num @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_num @ X8 ) ) ) ).

% infinite_growing
thf(fact_7291_infinite__growing,axiom,
    ! [X8: set_nat] :
      ( ( X8 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X8 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X8 )
                & ( ord_less_nat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X8 ) ) ) ).

% infinite_growing
thf(fact_7292_infinite__growing,axiom,
    ! [X8: set_int] :
      ( ( X8 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X8 )
           => ? [Xa2: int] :
                ( ( member_int @ Xa2 @ X8 )
                & ( ord_less_int @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_int @ X8 ) ) ) ).

% infinite_growing
thf(fact_7293_ex__min__if__finite,axiom,
    ! [S2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S2 )
            & ~ ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ S2 )
                  & ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7294_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S2 )
            & ~ ? [Xa2: real] :
                  ( ( member_real @ Xa2 @ S2 )
                  & ( ord_less_real @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7295_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( ( finite_finite_rat @ S2 )
     => ( ( S2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S2 )
            & ~ ? [Xa2: rat] :
                  ( ( member_rat @ Xa2 @ S2 )
                  & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7296_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( ( finite_finite_num @ S2 )
     => ( ( S2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S2 )
            & ~ ? [Xa2: num] :
                  ( ( member_num @ Xa2 @ S2 )
                  & ( ord_less_num @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7297_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S2 )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S2 )
                  & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7298_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S2 )
            & ~ ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ S2 )
                  & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7299_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_7300_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_7301_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_7302_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_7303_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_7304_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7305_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7306_finite__ranking__induct,axiom,
    ! [S2: set_Code_integer,P: set_Code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7307_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7308_finite__ranking__induct,axiom,
    ! [S2: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7309_finite__ranking__induct,axiom,
    ! [S2: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7310_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7311_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7312_finite__ranking__induct,axiom,
    ! [S2: set_Code_integer,P: set_Code_integer > $o,F: code_integer > num] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( 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 @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7313_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S6: set_nat] :
              ( ( finite_finite_nat @ S6 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_nat @ X3 @ S6 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7314_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B2: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ A8 )
                   => ( ord_le6747313008572928689nteger @ B2 @ X6 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7315_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B2: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X6: real] :
                    ( ( member_real @ X6 @ A8 )
                   => ( ord_less_real @ B2 @ X6 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7316_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B2: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X6: rat] :
                    ( ( member_rat @ X6 @ A8 )
                   => ( ord_less_rat @ B2 @ X6 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7317_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B2: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X6: num] :
                    ( ( member_num @ X6 @ A8 )
                   => ( ord_less_num @ B2 @ X6 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7318_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B2: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A8 )
                   => ( ord_less_nat @ B2 @ X6 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7319_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B2: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X6: int] :
                    ( ( member_int @ X6 @ A8 )
                   => ( ord_less_int @ B2 @ X6 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B2 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7320_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_7321_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_real,A2: real > real > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L1930518968523514909l_real @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7322_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_o,A2: real > $o > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L6234343332106409831real_o @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7323_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_nat,A2: real > nat > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_nat @ Li2 ) )
     => ( ( vEBT_L1446010312343316929al_nat @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7324_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_int,A2: real > int > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_int @ Li2 ) )
     => ( ( vEBT_L1443519841834266653al_int @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7325_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_real,A2: $o > real > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L4725278957065240257o_real @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7326_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_o,A2: $o > $o > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L7363604446928714179sn_o_o @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7327_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_nat,A2: $o > nat > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_nat @ Li2 ) )
     => ( ( vEBT_L4785011123346445925_o_nat @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7328_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_int,A2: $o > int > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_int @ Li2 ) )
     => ( ( vEBT_L4782520652837395649_o_int @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7329_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_real,A2: nat > real > assn] :
      ( ( ( size_size_list_nat @ L )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L6102073776069194049t_real @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7330_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_o,A2: nat > $o > assn] :
      ( ( ( size_size_list_nat @ L )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L7887682484454631235_nat_o @ A2 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_7331_accp__subset,axiom,
    ! [R1: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o,R22: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o] :
      ( ( ord_le1077754993875142464_nat_o @ R1 @ R22 )
     => ( ord_le7812727212727832188_nat_o @ ( accp_P2887432264394892906BT_nat @ R22 ) @ ( accp_P2887432264394892906BT_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7332_accp__subset,axiom,
    ! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
      ( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
     => ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).

% accp_subset
thf(fact_7333_accp__subset,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7334_accp__subset,axiom,
    ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_7335_accp__subset,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7336_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_7337_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_7338_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_7339_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_7340_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_7341_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_T2636463487746394924on_nat,A2: vEBT_VEBT > option_nat > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_7629718768684598413on_nat @ Q @ X2
        @ ^ [R: option_nat] : ( times_times_assn @ Q @ ( A2 @ Y2 @ R ) ) )
     => ( hoare_6480275734082232733on_nat @ Q @ ( vEBT_V792416675989592002on_nat @ N2 @ X2 )
        @ ^ [R: list_option_nat] : ( times_times_assn @ Q @ ( vEBT_L8010285020845282001on_nat @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) @ R ) ) ) ) ).

% repli_cons_repl
thf(fact_7342_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_Time_Heap_nat,A2: vEBT_VEBT > nat > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_3067605981109127869le_nat @ Q @ X2
        @ ^ [R: nat] : ( times_times_assn @ Q @ ( A2 @ Y2 @ R ) ) )
     => ( hoare_7964568885773372237st_nat @ Q @ ( vEBT_V7726092123322077554ei_nat @ N2 @ X2 )
        @ ^ [R: list_nat] : ( times_times_assn @ Q @ ( vEBT_L8296926524756676353BT_nat @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) @ R ) ) ) ) ).

% repli_cons_repl
thf(fact_7343_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_Time_Heap_o,A2: vEBT_VEBT > $o > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_hoare_triple_o @ Q @ X2
        @ ^ [R: $o] : ( times_times_assn @ Q @ ( A2 @ Y2 @ R ) ) )
     => ( hoare_9089481587091695345list_o @ Q @ ( vEBT_V2326993469660664182atei_o @ N2 @ X2 )
        @ ^ [R: list_o] : ( times_times_assn @ Q @ ( vEBT_L7489408758114837031VEBT_o @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) @ R ) ) ) ) ).

% repli_cons_repl
thf(fact_7344_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_T5738788834812785303t_unit,A2: vEBT_VEBT > product_unit > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_8945653483474564448t_unit @ Q @ X2
        @ ^ [R: product_unit] : ( times_times_assn @ Q @ ( A2 @ Y2 @ R ) ) )
     => ( hoare_8193881092815343078t_unit @ Q @ ( vEBT_V7483891112628345579t_unit @ N2 @ X2 )
        @ ^ [R: list_Product_unit] : ( times_times_assn @ Q @ ( vEBT_L8068554427805421084t_unit @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) @ R ) ) ) ) ).

% repli_cons_repl
thf(fact_7345_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_T8145700208782473153_VEBTi,A2: vEBT_VEBT > vEBT_VEBTi > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ Q @ X2
        @ ^ [R: vEBT_VEBTi] : ( times_times_assn @ Q @ ( A2 @ Y2 @ R ) ) )
     => ( hoare_3904069481286416050_VEBTi @ Q @ ( vEBT_V1859673955506687831_VEBTi @ N2 @ X2 )
        @ ^ [R: list_VEBT_VEBTi] : ( times_times_assn @ Q @ ( vEBT_L6296928887356842470_VEBTi @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) @ R ) ) ) ) ).

% repli_cons_repl
thf(fact_7346_repli__emp,axiom,
    ! [X2: heap_T2636463487746394924on_nat,A2: vEBT_VEBT > option_nat > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_7629718768684598413on_nat @ one_one_assn @ X2 @ ( A2 @ Y2 ) )
     => ( hoare_6480275734082232733on_nat @ one_one_assn @ ( vEBT_V792416675989592002on_nat @ N2 @ X2 ) @ ( vEBT_L8010285020845282001on_nat @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) ) ) ) ).

% repli_emp
thf(fact_7347_repli__emp,axiom,
    ! [X2: heap_Time_Heap_nat,A2: vEBT_VEBT > nat > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_3067605981109127869le_nat @ one_one_assn @ X2 @ ( A2 @ Y2 ) )
     => ( hoare_7964568885773372237st_nat @ one_one_assn @ ( vEBT_V7726092123322077554ei_nat @ N2 @ X2 ) @ ( vEBT_L8296926524756676353BT_nat @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) ) ) ) ).

% repli_emp
thf(fact_7348_repli__emp,axiom,
    ! [X2: heap_Time_Heap_o,A2: vEBT_VEBT > $o > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_hoare_triple_o @ one_one_assn @ X2 @ ( A2 @ Y2 ) )
     => ( hoare_9089481587091695345list_o @ one_one_assn @ ( vEBT_V2326993469660664182atei_o @ N2 @ X2 ) @ ( vEBT_L7489408758114837031VEBT_o @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) ) ) ) ).

% repli_emp
thf(fact_7349_repli__emp,axiom,
    ! [X2: heap_T5738788834812785303t_unit,A2: vEBT_VEBT > product_unit > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_8945653483474564448t_unit @ one_one_assn @ X2 @ ( A2 @ Y2 ) )
     => ( hoare_8193881092815343078t_unit @ one_one_assn @ ( vEBT_V7483891112628345579t_unit @ N2 @ X2 ) @ ( vEBT_L8068554427805421084t_unit @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) ) ) ) ).

% repli_emp
thf(fact_7350_repli__emp,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,A2: vEBT_VEBT > vEBT_VEBTi > assn,Y2: vEBT_VEBT,N2: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ one_one_assn @ X2 @ ( A2 @ Y2 ) )
     => ( hoare_3904069481286416050_VEBTi @ one_one_assn @ ( vEBT_V1859673955506687831_VEBTi @ N2 @ X2 ) @ ( vEBT_L6296928887356842470_VEBTi @ A2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) ) ) ) ).

% repli_emp
thf(fact_7351_length__replicate,axiom,
    ! [N2: nat,X2: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_7352_length__replicate,axiom,
    ! [N2: nat,X2: real] :
      ( ( size_size_list_real @ ( replicate_real @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_7353_length__replicate,axiom,
    ! [N2: nat,X2: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_7354_length__replicate,axiom,
    ! [N2: nat,X2: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_7355_length__replicate,axiom,
    ! [N2: nat,X2: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_7356_Ball__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7357_Ball__set__replicate,axiom,
    ! [N2: nat,A: real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7358_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7359_Ball__set__replicate,axiom,
    ! [N2: nat,A: produc6575502325842934193n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ! [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7360_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7361_Bex__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7362_Bex__set__replicate,axiom,
    ! [N2: nat,A: real,P: real > $o] :
      ( ( ? [X: real] :
            ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7363_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ? [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7364_Bex__set__replicate,axiom,
    ! [N2: nat,A: produc6575502325842934193n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ? [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7365_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7366_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_7367_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_7368_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_7369_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_7370_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_7371_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_7372_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_7373_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_7374_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_7375_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_7376_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_7377_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_7378_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_7379_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_7380_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_real] :
      ( ( map_Pr6195879527588727455T_VEBT @ produc8110914911036349469T_real @ ( map_re8618229306769252225T_real @ ( produc8117437818029410057T_real @ K ) @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7381_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_o] :
      ( ( map_Pr4868735216952053677T_VEBT @ produc4993121158135996263VEBT_o @ ( map_o_6754667662019005495VEBT_o @ ( produc8721562602347293563VEBT_o @ K ) @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7382_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_nat] :
      ( ( map_Pr1380729192516676091T_VEBT @ produc8713918199166443969BT_nat @ ( map_na4631810538828370761BT_nat @ ( produc738532404422230701BT_nat @ K ) @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7383_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_int] :
      ( ( map_Pr3257657825534036127T_VEBT @ produc8711427728657393693BT_int @ ( map_in8151279748432256513BT_int @ ( produc736041933913180425BT_int @ K ) @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_int @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7384_map__fst__mk__fst,axiom,
    ! [K: assn,L: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ ( map_as2373307505041272643n_assn @ ( produc118845697133431529n_assn @ K ) @ L ) )
      = ( replicate_assn @ ( size_size_list_assn @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7385_map__fst__mk__fst,axiom,
    ! [K: uint32,L: list_uint32] :
      ( ( map_Pr2530660914518622561uint32 @ produc9004433772639906525uint32 @ ( map_ui5747794299746474503uint32 @ ( produc1400373151660368625uint32 @ K ) @ L ) )
      = ( replicate_uint32 @ ( size_s4844771616002835472uint32 @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7386_map__fst__mk__fst,axiom,
    ! [K: nat,L: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ K ) @ L ) )
      = ( replicate_nat @ ( size_size_list_nat @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7387_map__fst__mk__fst,axiom,
    ! [K: int,L: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ ( map_in7157766398909135175nt_int @ ( product_Pair_int_int @ K ) @ L ) )
      = ( replicate_int @ ( size_size_list_int @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7388_map__fst__mk__fst,axiom,
    ! [K: heap_e7401611519738050253t_unit,L: list_set_nat] :
      ( ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat @ ( map_se5435525824004667726et_nat @ ( produc7507926704131184380et_nat @ K ) @ L ) )
      = ( replic5388364368018022029t_unit @ ( size_s3254054031482475050et_nat @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7389_map__fst__mk__fst,axiom,
    ! [K: int > option6357759511663192854e_term,L: list_P5707943133018811711nt_int] :
      ( ( map_Pr7726103239032798900e_term @ produc6230002227079971283nt_int @ ( map_Pr1306541819098601986nt_int @ ( produc4305682042979456191nt_int @ K ) @ L ) )
      = ( replic6271474913745183777e_term @ ( size_s5157815400016825771nt_int @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_7390_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_real] :
      ( ( map_Pr6147841162850987569T_VEBT @ produc5083336317046741121T_VEBT
        @ ( map_re7205069664741861231T_VEBT
          @ ^ [X: real] : ( produc6931449550656315951T_VEBT @ X @ K )
          @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7391_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_o] :
      ( ( map_Pr7652832201708611317T_VEBT @ produc7938581201502569057T_VEBT
        @ ( map_o_8925299737569714927T_VEBT
          @ ^ [X: $o] : ( produc2982872950893828659T_VEBT @ X @ K )
          @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7392_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_nat] :
      ( ( map_Pr8570210702748812117T_VEBT @ produc8172668247895388509T_VEBT
        @ ( map_na3584885621601055599T_VEBT
          @ ^ [X: nat] : ( produc599794634098209291T_VEBT @ X @ K )
          @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7393_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_int] :
      ( ( map_Pr1314269154781486001T_VEBT @ produc1678900780639429121T_VEBT
        @ ( map_in4788438383458178671T_VEBT
          @ ^ [X: int] : ( produc3329399203697025711T_VEBT @ X @ K )
          @ L ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_int @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7394_map__snd__mk__snd,axiom,
    ! [K: nat,L: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ K )
          @ L ) )
      = ( replicate_nat @ ( size_size_list_nat @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7395_map__snd__mk__snd,axiom,
    ! [K: int,L: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int
        @ ( map_in7157766398909135175nt_int
          @ ^ [X: int] : ( product_Pair_int_int @ X @ K )
          @ L ) )
      = ( replicate_int @ ( size_size_list_int @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7396_map__snd__mk__snd,axiom,
    ! [K: code_integer,L: list_Code_integer] :
      ( ( map_Pr1857711230949937460nteger @ produc6174133586879617921nteger
        @ ( map_Co3589949550033412536nteger
          @ ^ [X: code_integer] : ( produc1086072967326762835nteger @ X @ K )
          @ L ) )
      = ( replic7707675349574490269nteger @ ( size_s3445333598471063425nteger @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7397_map__snd__mk__snd,axiom,
    ! [K: assn,L: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn
        @ ( map_as2373307505041272643n_assn
          @ ^ [X: assn] : ( produc118845697133431529n_assn @ X @ K )
          @ L ) )
      = ( replicate_assn @ ( size_size_list_assn @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7398_map__snd__mk__snd,axiom,
    ! [K: set_nat,L: list_h2745205591237434579t_unit] :
      ( ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat
        @ ( map_he4567745686773187723et_nat
          @ ^ [X: heap_e7401611519738050253t_unit] : ( produc7507926704131184380et_nat @ X @ K )
          @ L ) )
      = ( replicate_set_nat @ ( size_s7011499401410611007t_unit @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7399_map__snd__mk__snd,axiom,
    ! [K: product_prod_int_int,L: list_i8448526496819171953e_term] :
      ( ( map_Pr5958523780849169702nt_int @ produc3162348030201620241nt_int
        @ ( map_in8886716077063074740nt_int
          @ ^ [X: int > option6357759511663192854e_term] : ( produc4305682042979456191nt_int @ X @ K )
          @ L ) )
      = ( replic1057375728873637753nt_int @ ( size_s8881356780637927685e_term @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_7400_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_7401_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_7402_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_7403_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_7404_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_7405_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_7406_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_7407_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_7408_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_7409_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_7410_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_7411_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_7412_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_7413_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_7414_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_7415_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_7416_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_7417_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_7418_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_7419_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_7420_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_7421_set__replicate__Suc,axiom,
    ! [N2: nat,X2: produc6575502325842934193n_assn] :
      ( ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ ( suc @ N2 ) @ X2 ) )
      = ( insert5290817439147925377n_assn @ X2 @ bot_bo1176836662018730877n_assn ) ) ).

% set_replicate_Suc
thf(fact_7422_set__replicate__Suc,axiom,
    ! [N2: nat,X2: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N2 ) @ X2 ) )
      = ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_7423_set__replicate__Suc,axiom,
    ! [N2: nat,X2: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N2 ) @ X2 ) )
      = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_7424_set__replicate__Suc,axiom,
    ! [N2: nat,X2: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N2 ) @ X2 ) )
      = ( insert_int @ X2 @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_7425_set__replicate__Suc,axiom,
    ! [N2: nat,X2: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N2 ) @ X2 ) )
      = ( insert_real @ X2 @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_7426_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_7427_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_7428_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_7429_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_7430_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_7431_list__assn__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBT > assn,A9: vEBT_VEBT > vEBT_VEBT > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_VEBT_VEBT @ Xi @ ( set_VEBT_VEBT2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L1279224858307276611T_VEBT @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L1279224858307276611T_VEBT @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7432_list__assn__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: vEBT_VEBT > nat > assn,A9: vEBT_VEBT > nat > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: nat] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8296926524756676353BT_nat @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L8296926524756676353BT_nat @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7433_list__assn__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: vEBT_VEBT > real > assn,A9: vEBT_VEBT > real > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: real] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_real @ Xi @ ( set_real2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L5781919052683127133T_real @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L5781919052683127133T_real @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7434_list__assn__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A2: vEBT_VEBT > int > assn,A9: vEBT_VEBT > int > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: int] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_int @ Xi @ ( set_int2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8294436054247626077BT_int @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L8294436054247626077BT_int @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7435_list__assn__cong,axiom,
    ! [Xs2: list_nat,Xs4: list_nat,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: nat > vEBT_VEBT > assn,A9: nat > vEBT_VEBT > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: vEBT_VEBT] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member_VEBT_VEBT @ Xi @ ( set_VEBT_VEBT2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8158188754432654943T_VEBT @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L8158188754432654943T_VEBT @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7436_list__assn__cong,axiom,
    ! [Xs2: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat,A2: nat > nat > assn,A9: nat > nat > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8301102511889123557at_nat @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L8301102511889123557at_nat @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7437_list__assn__cong,axiom,
    ! [Xs2: list_nat,Xs4: list_nat,Xsi: list_real,Xsi2: list_real,A2: nat > real > assn,A9: nat > real > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: real] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member_real @ Xi @ ( set_real2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L6102073776069194049t_real @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L6102073776069194049t_real @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7438_list__assn__cong,axiom,
    ! [Xs2: list_nat,Xs4: list_nat,Xsi: list_int,Xsi2: list_int,A2: nat > int > assn,A9: nat > int > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: int] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member_int @ Xi @ ( set_int2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8298612041380073281at_int @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L8298612041380073281at_int @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7439_list__assn__cong,axiom,
    ! [Xs2: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: real > vEBT_VEBT > assn,A9: real > vEBT_VEBT > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: real,Xi: vEBT_VEBT] :
              ( ( member_real @ X3 @ ( set_real2 @ Xs4 ) )
             => ( ( member_VEBT_VEBT @ Xi @ ( set_VEBT_VEBT2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L4595930785310033027T_VEBT @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L4595930785310033027T_VEBT @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7440_list__assn__cong,axiom,
    ! [Xs2: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat,A2: real > nat > assn,A9: real > nat > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: real,Xi: nat] :
              ( ( member_real @ X3 @ ( set_real2 @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A9 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L1446010312343316929al_nat @ A2 @ Xs2 @ Xsi )
            = ( vEBT_L1446010312343316929al_nat @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_7441_accp__subset__induct,axiom,
    ! [D5: produc9072475918466114483BT_nat > $o,R2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o,X2: produc9072475918466114483BT_nat,P: produc9072475918466114483BT_nat > $o] :
      ( ( ord_le7812727212727832188_nat_o @ D5 @ ( accp_P2887432264394892906BT_nat @ R2 ) )
     => ( ! [X3: produc9072475918466114483BT_nat,Z3: produc9072475918466114483BT_nat] :
            ( ( D5 @ X3 )
           => ( ( R2 @ Z3 @ X3 )
             => ( D5 @ Z3 ) ) )
       => ( ( D5 @ X2 )
         => ( ! [X3: produc9072475918466114483BT_nat] :
                ( ( D5 @ X3 )
               => ( ! [Z5: produc9072475918466114483BT_nat] :
                      ( ( R2 @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7442_accp__subset__induct,axiom,
    ! [D5: product_prod_num_num > $o,R2: product_prod_num_num > product_prod_num_num > $o,X2: product_prod_num_num,P: product_prod_num_num > $o] :
      ( ( ord_le2239182809043710856_num_o @ D5 @ ( accp_P3113834385874906142um_num @ R2 ) )
     => ( ! [X3: product_prod_num_num,Z3: product_prod_num_num] :
            ( ( D5 @ X3 )
           => ( ( R2 @ Z3 @ X3 )
             => ( D5 @ Z3 ) ) )
       => ( ( D5 @ X2 )
         => ( ! [X3: product_prod_num_num] :
                ( ( D5 @ X3 )
               => ( ! [Z5: product_prod_num_num] :
                      ( ( R2 @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7443_accp__subset__induct,axiom,
    ! [D5: product_prod_nat_nat > $o,R2: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D5 @ ( accp_P4275260045618599050at_nat @ R2 ) )
     => ( ! [X3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( ( D5 @ X3 )
           => ( ( R2 @ Z3 @ X3 )
             => ( D5 @ Z3 ) ) )
       => ( ( D5 @ X2 )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( D5 @ X3 )
               => ( ! [Z5: product_prod_nat_nat] :
                      ( ( R2 @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7444_accp__subset__induct,axiom,
    ! [D5: product_prod_int_int > $o,R2: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D5 @ ( accp_P1096762738010456898nt_int @ R2 ) )
     => ( ! [X3: product_prod_int_int,Z3: product_prod_int_int] :
            ( ( D5 @ X3 )
           => ( ( R2 @ Z3 @ X3 )
             => ( D5 @ Z3 ) ) )
       => ( ( D5 @ X2 )
         => ( ! [X3: product_prod_int_int] :
                ( ( D5 @ X3 )
               => ( ! [Z5: product_prod_int_int] :
                      ( ( R2 @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7445_accp__subset__induct,axiom,
    ! [D5: nat > $o,R2: nat > nat > $o,X2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat_o @ D5 @ ( accp_nat @ R2 ) )
     => ( ! [X3: nat,Z3: nat] :
            ( ( D5 @ X3 )
           => ( ( R2 @ Z3 @ X3 )
             => ( D5 @ Z3 ) ) )
       => ( ( D5 @ X2 )
         => ( ! [X3: nat] :
                ( ( D5 @ X3 )
               => ( ! [Z5: nat] :
                      ( ( R2 @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7446_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_7447_extract__pre__list__assn__lengthD,axiom,
    ! [A2: real > real > assn,Xs2: list_real,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1930518968523514909l_real @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_real @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7448_extract__pre__list__assn__lengthD,axiom,
    ! [A2: $o > real > assn,Xs2: list_o,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4725278957065240257o_real @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_o @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7449_extract__pre__list__assn__lengthD,axiom,
    ! [A2: nat > real > assn,Xs2: list_nat,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6102073776069194049t_real @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_nat @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7450_extract__pre__list__assn__lengthD,axiom,
    ! [A2: int > real > assn,Xs2: list_int,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L8288995350762215837t_real @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_int @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7451_extract__pre__list__assn__lengthD,axiom,
    ! [A2: real > $o > assn,Xs2: list_real,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6234343332106409831real_o @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_real @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7452_extract__pre__list__assn__lengthD,axiom,
    ! [A2: $o > $o > assn,Xs2: list_o,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L7363604446928714179sn_o_o @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_o @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7453_extract__pre__list__assn__lengthD,axiom,
    ! [A2: nat > $o > assn,Xs2: list_nat,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L7887682484454631235_nat_o @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_nat @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7454_extract__pre__list__assn__lengthD,axiom,
    ! [A2: int > $o > assn,Xs2: list_int,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6066640139021943271_int_o @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_int @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7455_extract__pre__list__assn__lengthD,axiom,
    ! [A2: real > nat > assn,Xs2: list_real,Xsi: list_nat,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1446010312343316929al_nat @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_nat @ Xsi )
        = ( size_size_list_real @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7456_extract__pre__list__assn__lengthD,axiom,
    ! [A2: $o > nat > assn,Xs2: list_o,Xsi: list_nat,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4785011123346445925_o_nat @ A2 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_nat @ Xsi )
        = ( size_size_list_o @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_7457_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_7458_VEBT__internal_Ospace_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space.pelims
thf(fact_7459_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space2 @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
thf(fact_7460_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 ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xa
                    = ( vEBT_Leafi @ Ai @ Bi ) )
                 => ( ( Y2
                      = ( pure_assn
                        @ ( ( Ai = A3 )
                          & ( Bi = B2 ) ) ) )
                   => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B2 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
         => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
               => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                    ( ( Xa
                      = ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
                   => ( ( Y2
                        = ( times_times_assn
                          @ ( times_times_assn
                            @ ( pure_assn
                              @ ( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ) )
                            @ ( vEBT_vebt_assn_raw @ Summary2 @ 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 ) ) ) ) )
                     => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) ) ) ) )
           => ( ! [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_7461_vebt__maxti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leafi @ A3 @ B2 ) )
             => ( ( ( B2
                   => ( Y2
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                  & ( ~ B2
                   => ( ( A3
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                      & ( ~ A3
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A3 @ B2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxti.pelims
thf(fact_7462_vebt__minti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leafi @ A3 @ B2 ) )
             => ( ( ( A3
                   => ( Y2
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                  & ( ~ A3
                   => ( ( B2
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                      & ( ~ B2
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A3 @ B2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_minti.pelims
thf(fact_7463_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: real] :
      ( ( ( vEBT_VEBT_cnt @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2 = one_one_real )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
thf(fact_7464_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2 = one_one_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
thf(fact_7465_vebt__maxt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( ( B2
                   => ( Y2
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B2
                   => ( ( A3
                       => ( Y2
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y2 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_7466_vebt__mint_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( ( A3
                   => ( Y2
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B2
                       => ( Y2
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B2
                       => ( Y2 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_7467_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 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_7468_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 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_7469_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 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leafi @ Uu2 @ $true ) )
                 => ( ( Y2
                      = ( heap_Time_return_o @ $false ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( ( X2
                      = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( ( Y2
                        = ( heap_Time_return_o @ $true ) )
                     => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( ( X2
                        = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                     => ( ( Y2
                          = ( heap_Time_return_o @ $false ) )
                       => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
thf(fact_7470_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 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_7471_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat,Acc: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
            = Acc ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_7472_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X2: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y2: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa @ Xb @ Xc )
        = Y2 )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y2 = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y2
                  = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_7473_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X2: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,Xa: option2621746655072343315it_nat,Xb: option2621746655072343315it_nat,Y2: option2621746655072343315it_nat] :
      ( ( ( vEBT_V819568868292977612it_nat @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp_P6139839302380951924it_nat @ vEBT_V3802522976469930835it_nat @ ( produc8579712001971957723it_nat @ X2 @ ( produc6851560022941992023it_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_P1551326421579882414it_nat )
           => ( ( Y2 = none_P1551326421579882414it_nat )
             => ~ ( accp_P6139839302380951924it_nat @ vEBT_V3802522976469930835it_nat @ ( produc8579712001971957723it_nat @ X2 @ ( produc6851560022941992023it_nat @ none_P1551326421579882414it_nat @ Xb ) ) ) ) )
         => ( ! [V2: produc120671012495760973it_nat] :
                ( ( Xa
                  = ( some_P2407035485129114418it_nat @ V2 ) )
               => ( ( Xb = none_P1551326421579882414it_nat )
                 => ( ( Y2 = none_P1551326421579882414it_nat )
                   => ~ ( accp_P6139839302380951924it_nat @ vEBT_V3802522976469930835it_nat @ ( produc8579712001971957723it_nat @ X2 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ V2 ) @ none_P1551326421579882414it_nat ) ) ) ) ) )
           => ~ ! [A3: produc120671012495760973it_nat] :
                  ( ( Xa
                    = ( some_P2407035485129114418it_nat @ A3 ) )
                 => ! [B2: produc120671012495760973it_nat] :
                      ( ( Xb
                        = ( some_P2407035485129114418it_nat @ B2 ) )
                     => ( ( Y2
                          = ( some_P2407035485129114418it_nat @ ( X2 @ A3 @ B2 ) ) )
                       => ~ ( accp_P6139839302380951924it_nat @ vEBT_V3802522976469930835it_nat @ ( produc8579712001971957723it_nat @ X2 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ A3 ) @ ( some_P2407035485129114418it_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7474_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,Xa: option7339022715339332451it_nat,Xb: option7339022715339332451it_nat,Y2: option7339022715339332451it_nat] :
      ( ( ( vEBT_V613753007643960916it_nat @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp_P4085165796030815644it_nat @ vEBT_V482137685244371085it_nat @ ( produc2320005133921938071it_nat @ X2 @ ( produc9206348758962449759it_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_P7668321371905463026it_nat )
           => ( ( Y2 = none_P7668321371905463026it_nat )
             => ~ ( accp_P4085165796030815644it_nat @ vEBT_V482137685244371085it_nat @ ( produc2320005133921938071it_nat @ X2 @ ( produc9206348758962449759it_nat @ none_P7668321371905463026it_nat @ Xb ) ) ) ) )
         => ( ! [V2: produc8047831477865546771it_nat] :
                ( ( Xa
                  = ( some_P468703482102919278it_nat @ V2 ) )
               => ( ( Xb = none_P7668321371905463026it_nat )
                 => ( ( Y2 = none_P7668321371905463026it_nat )
                   => ~ ( accp_P4085165796030815644it_nat @ vEBT_V482137685244371085it_nat @ ( produc2320005133921938071it_nat @ X2 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ V2 ) @ none_P7668321371905463026it_nat ) ) ) ) ) )
           => ~ ! [A3: produc8047831477865546771it_nat] :
                  ( ( Xa
                    = ( some_P468703482102919278it_nat @ A3 ) )
                 => ! [B2: produc8047831477865546771it_nat] :
                      ( ( Xb
                        = ( some_P468703482102919278it_nat @ B2 ) )
                     => ( ( Y2
                          = ( some_P468703482102919278it_nat @ ( X2 @ A3 @ B2 ) ) )
                       => ~ ( accp_P4085165796030815644it_nat @ vEBT_V482137685244371085it_nat @ ( produc2320005133921938071it_nat @ X2 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ A3 ) @ ( some_P468703482102919278it_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7475_VEBT__internal_Ooption__shift_Opelims,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 )
     => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_P5556105721700978146at_nat )
           => ( ( Y2 = none_P5556105721700978146at_nat )
             => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xa
                  = ( some_P7363390416028606310at_nat @ V2 ) )
               => ( ( Xb = none_P5556105721700978146at_nat )
                 => ( ( Y2 = none_P5556105721700978146at_nat )
                   => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
           => ~ ! [A3: product_prod_nat_nat] :
                  ( ( Xa
                    = ( some_P7363390416028606310at_nat @ A3 ) )
                 => ! [B2: product_prod_nat_nat] :
                      ( ( Xb
                        = ( some_P7363390416028606310at_nat @ B2 ) )
                     => ( ( Y2
                          = ( some_P7363390416028606310at_nat @ ( X2 @ A3 @ B2 ) ) )
                       => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7476_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X2: num > num > num,Xa: option_num,Xb: option_num,Y2: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_num )
           => ( ( Y2 = none_num )
             => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
         => ( ! [V2: num] :
                ( ( Xa
                  = ( some_num @ V2 ) )
               => ( ( Xb = none_num )
                 => ( ( Y2 = none_num )
                   => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
           => ~ ! [A3: num] :
                  ( ( Xa
                    = ( some_num @ A3 ) )
                 => ! [B2: num] :
                      ( ( Xb
                        = ( some_num @ B2 ) )
                     => ( ( Y2
                          = ( some_num @ ( X2 @ A3 @ B2 ) ) )
                       => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7477_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X2: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y2: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_nat )
           => ( ( Y2 = none_nat )
             => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
         => ( ! [V2: nat] :
                ( ( Xa
                  = ( some_nat @ V2 ) )
               => ( ( Xb = none_nat )
                 => ( ( Y2 = none_nat )
                   => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
           => ~ ! [A3: nat] :
                  ( ( Xa
                    = ( some_nat @ A3 ) )
                 => ! [B2: nat] :
                      ( ( Xb
                        = ( some_nat @ B2 ) )
                     => ( ( Y2
                          = ( some_nat @ ( X2 @ A3 @ B2 ) ) )
                       => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7478_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X2: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y2: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa @ Xb @ Xc )
        = Y2 )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y2 = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y2
            = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_7479_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F4: nat > nat > nat,A4: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F4 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F4 @ A4 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_7480_freeze__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( hoare_3904069481286416050_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_8141364883501958055_VEBTi @ A )
      @ ^ [R: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( pure_assn @ ( R = Xs2 ) ) ) ) ).

% freeze_rule
thf(fact_7481_in__finite__psubset,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B5 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A2 @ B5 )
        & ( finite_finite_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_7482_in__finite__psubset,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A2 @ B5 ) @ finite_psubset_int )
      = ( ( ord_less_set_int @ A2 @ B5 )
        & ( finite_finite_int @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_7483_in__finite__psubset,axiom,
    ! [A2: set_complex,B5: set_complex] :
      ( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A2 @ B5 ) @ finite8643634255014194347omplex )
      = ( ( ord_less_set_complex @ A2 @ B5 )
        & ( finite3207457112153483333omplex @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_7484_in__finite__psubset,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer] :
      ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A2 @ B5 ) @ finite2416775604798480986nteger )
      = ( ( ord_le1307284697595431911nteger @ A2 @ B5 )
        & ( finite6017078050557962740nteger @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_7485_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > real,Y2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( times_times_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7486_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > real,Y2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( times_times_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7487_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > real,Y2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( times_times_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7488_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > real,Y2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( times_times_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7489_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X2: complex > real,Y2: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( times_times_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7490_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X2: code_integer > real,Y2: code_integer > real] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
                & ( ( times_times_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7491_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > rat,Y2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_rat ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_rat ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( times_times_rat @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7492_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > rat,Y2: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( times_times_rat @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7493_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > rat,Y2: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( times_times_rat @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7494_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > rat,Y2: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( times_times_rat @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7495_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > uint32,Y2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( plus_plus_uint32 @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7496_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > uint32,Y2: real > uint32] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( plus_plus_uint32 @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7497_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > uint32,Y2: nat > uint32] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( plus_plus_uint32 @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7498_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > uint32,Y2: int > uint32] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( plus_plus_uint32 @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7499_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X2: complex > uint32,Y2: complex > uint32] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( plus_plus_uint32 @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7500_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X2: code_integer > uint32,Y2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
                & ( ( plus_plus_uint32 @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7501_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > real,Y2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7502_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > real,Y2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7503_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > real,Y2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7504_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > real,Y2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X2 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y2 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7505_ceiling__log__eq__powr__iff,axiom,
    ! [X2: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X2 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
            & ( ord_less_eq_real @ X2 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_7506_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_7507_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_7508_powr__nonneg__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X2 ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_7509_powr__less__cancel__iff,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_7510_powr__eq__one__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X2 )
          = one_one_real )
        = ( X2 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_7511_powr__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ one_one_real )
        = X2 ) ) ).

% powr_one
thf(fact_7512_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_7513_powr__le__cancel__iff,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_7514_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_7515_log__powr__cancel,axiom,
    ! [A: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y2 ) )
          = Y2 ) ) ) ).

% log_powr_cancel
thf(fact_7516_powr__log__cancel,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( powr_real @ A @ ( log @ A @ X2 ) )
            = X2 ) ) ) ) ).

% powr_log_cancel
thf(fact_7517_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_7518_powr__powr,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X2 @ A ) @ B )
      = ( powr_real @ X2 @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_7519_powr__ge__pzero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y2 ) ) ).

% powr_ge_pzero
thf(fact_7520_powr__mono2,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y2 )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_7521_powr__less__mono,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).

% powr_less_mono
thf(fact_7522_powr__less__cancel,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_7523_powr__mono,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X2 )
       => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).

% powr_mono
thf(fact_7524_powr__mono2_H,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y2 )
         => ( ord_less_eq_real @ ( powr_real @ Y2 @ A ) @ ( powr_real @ X2 @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_7525_powr__less__mono2,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ Y2 )
         => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_7526_powr__inj,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X2 )
            = ( powr_real @ A @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% powr_inj
thf(fact_7527_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_7528_powr__le1,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_7529_powr__mono__both,axiom,
    ! [A: real,B: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X2 )
         => ( ( ord_less_eq_real @ X2 @ Y2 )
           => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_7530_ge__one__powr__ge__zero,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_7531_powr__divide,axiom,
    ! [X2: real,Y2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( powr_real @ ( divide_divide_real @ X2 @ Y2 ) @ A )
          = ( divide_divide_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_divide
thf(fact_7532_powr__mult,axiom,
    ! [X2: real,Y2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( powr_real @ ( times_times_real @ X2 @ Y2 ) @ A )
          = ( times_times_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_mult
thf(fact_7533_log__base__powr,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X2 )
        = ( divide_divide_real @ ( log @ A @ X2 ) @ B ) ) ) ).

% log_base_powr
thf(fact_7534_log__powr,axiom,
    ! [X2: real,B: real,Y2: real] :
      ( ( X2 != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X2 @ Y2 ) )
        = ( times_times_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ).

% log_powr
thf(fact_7535_powr__add,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( powr_real @ X2 @ ( plus_plus_real @ A @ B ) )
      = ( times_times_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ).

% powr_add
thf(fact_7536_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_7537_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_7538_less__log__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ Y2 @ ( log @ B @ X2 ) )
          = ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ).

% less_log_iff
thf(fact_7539_log__less__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ B @ X2 ) @ Y2 )
          = ( ord_less_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ).

% log_less_iff
thf(fact_7540_less__powr__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ ( powr_real @ B @ Y2 ) )
          = ( ord_less_real @ ( log @ B @ X2 ) @ Y2 ) ) ) ) ).

% less_powr_iff
thf(fact_7541_powr__less__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X2 )
          = ( ord_less_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ) ).

% powr_less_iff
thf(fact_7542_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_7543_le__log__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ Y2 @ ( log @ B @ X2 ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ).

% le_log_iff
thf(fact_7544_log__le__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y2 )
          = ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ).

% log_le_iff
thf(fact_7545_le__powr__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y2 ) )
          = ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y2 ) ) ) ) ).

% le_powr_iff
thf(fact_7546_powr__le__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X2 )
          = ( ord_less_eq_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ) ).

% powr_le_iff
thf(fact_7547_log__add__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ ( log @ B @ X2 ) @ Y2 )
            = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_7548_add__log__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ Y2 @ ( log @ B @ X2 ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_7549_minus__log__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ Y2 @ ( log @ B @ X2 ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_7550_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A6: set_nat,B6: set_nat] :
            ( ( ord_less_set_nat @ A6 @ B6 )
            & ( finite_finite_nat @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_7551_finite__psubset__def,axiom,
    ( finite_psubset_int
    = ( collec957716948307931664et_int
      @ ( produc4109468873575309990_int_o
        @ ^ [A6: set_int,B6: set_int] :
            ( ( ord_less_set_int @ A6 @ B6 )
            & ( finite_finite_int @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_7552_finite__psubset__def,axiom,
    ( finite8643634255014194347omplex
    = ( collec5108298041176329748omplex
      @ ( produc3914248068834153634plex_o
        @ ^ [A6: set_complex,B6: set_complex] :
            ( ( ord_less_set_complex @ A6 @ B6 )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_7553_finite__psubset__def,axiom,
    ( finite2416775604798480986nteger
    = ( collec2770208431294612722nteger
      @ ( produc1658495936642081476eger_o
        @ ^ [A6: set_Code_integer,B6: set_Code_integer] :
            ( ( ord_le1307284697595431911nteger @ A6 @ B6 )
            & ( finite6017078050557962740nteger @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_7554_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_7555_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_7556_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8115118780965096967l_num1 @ one_on7727431528512463931l_num1 )
    = ( numera7442385471795722001l_num1 @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7557_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7558_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7559_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7560_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_7561_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_7562_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
      = zero_zero_nat ) ).

% VEBT_internal.height.simps(1)
thf(fact_7563_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_7564_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_7565_sum_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A2 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_7566_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_7567_of__nat__sum,axiom,
    ! [F: complex > nat,A2: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( semiri8010041392384452111omplex @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7568_of__nat__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7569_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7570_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri4939895301339042750nteger @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups7501900531339628137nteger
        @ ^ [X: nat] : ( semiri4939895301339042750nteger @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7571_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7572_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7573_of__int__sum,axiom,
    ! [F: complex > int,A2: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( ring_17405671764205052669omplex @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_7574_of__int__sum,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_7575_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups8778361861064173332t_real
        @ ^ [X: int] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_7576_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups3906332499630173760nt_rat
        @ ^ [X: int] : ( ring_1_of_int_rat @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_7577_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( ring_1_of_int_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_7578_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K4: vEBT_VEBT] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K4: vEBT_VEBT] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_7579_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > uint32] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K4: real] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K4: real] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_7580_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > uint32] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups833757482993574392uint32
              @ ^ [K4: nat] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups833757482993574392uint32
              @ ^ [K4: nat] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_7581_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > uint32] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K4: int] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K4: int] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_7582_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > uint32] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K4: complex] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K4: complex] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_7583_sum_Odelta_H,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K4: code_integer] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K4: code_integer] : ( if_uint32 @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_7584_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7585_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K4: real] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K4: real] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7586_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K4: int] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K4: int] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7587_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K4: complex] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K4: complex] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7588_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K4: vEBT_VEBT] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K4: vEBT_VEBT] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_7589_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > uint32] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K4: real] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K4: real] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_7590_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > uint32] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups833757482993574392uint32
              @ ^ [K4: nat] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups833757482993574392uint32
              @ ^ [K4: nat] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_7591_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > uint32] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K4: int] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K4: int] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_7592_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > uint32] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K4: complex] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K4: complex] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_7593_sum_Odelta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K4: code_integer] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K4: code_integer] : ( if_uint32 @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_uint32 )
              @ S2 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_7594_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7595_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K4: real] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K4: real] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7596_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K4: int] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K4: int] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7597_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K4: complex] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K4: complex] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7598_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8115118780965096967l_num1 @ zero_z3563351764282998399l_num1 )
    = one_on7727431528512463931l_num1 ) ).

% dbl_inc_simps(2)
thf(fact_7599_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu4269007558841261821uint32 @ zero_zero_uint32 )
    = one_one_uint32 ) ).

% dbl_inc_simps(2)
thf(fact_7600_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_7601_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_7602_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_7603_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8115118780965096967l_num1 @ ( numera7442385471795722001l_num1 @ K ) )
      = ( numera7442385471795722001l_num1 @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7604_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7605_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7606_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7607_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7608_sum_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X2 @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X2 @ A2 ) )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7609_sum_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X2 @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X2 @ A2 ) )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7610_sum_Oinsert,axiom,
    ! [A2: set_complex,X2: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X2 @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X2 @ A2 ) )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7611_sum_Oinsert,axiom,
    ! [A2: set_Code_integer,X2: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X2 @ A2 )
       => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X2 @ A2 ) )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7612_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7613_sum_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X2 @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X2 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7614_sum_Oinsert,axiom,
    ! [A2: set_nat,X2: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X2 @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X2 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7615_sum_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X2 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X2 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7616_sum_Oinsert,axiom,
    ! [A2: set_complex,X2: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X2 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X2 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7617_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X2: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
        @ X2 )
      = ( ( A @ zero_zero_nat )
        = X2 ) ) ).

% powser_sums_zero_iff
thf(fact_7618_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > uint32] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups833757482993574392uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_uint32 ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups833757482993574392uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_uint32 @ ( groups833757482993574392uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7619_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_7620_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_7621_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_7622_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_7623_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_7624_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_7625_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_7626_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_7627_sum_Oswap,axiom,
    ! [G: nat > nat > nat,B5: set_nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I2: nat] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% sum.swap
thf(fact_7628_sum_Oswap,axiom,
    ! [G: complex > complex > complex,B5: set_complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [I2: complex] : ( groups7754918857620584856omplex @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups7754918857620584856omplex
        @ ^ [J3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [I2: complex] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% sum.swap
thf(fact_7629_sum_Oswap,axiom,
    ! [G: int > int > int,B5: set_int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [I2: int] : ( groups4538972089207619220nt_int @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups4538972089207619220nt_int
        @ ^ [J3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [I2: int] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% sum.swap
thf(fact_7630_sum_Oswap,axiom,
    ! [G: nat > nat > real,B5: set_nat,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% sum.swap
thf(fact_7631_norm__sum,axiom,
    ! [F: nat > complex,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( real_V1022390504157884413omplex @ ( F @ I2 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_7632_norm__sum,axiom,
    ! [F: complex > complex,A2: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I2: complex] : ( real_V1022390504157884413omplex @ ( F @ I2 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_7633_norm__sum,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( real_V7735802525324610683m_real @ ( F @ I2 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_7634_sum__norm__le,axiom,
    ! [S2: set_VEBT_VEBT,F: vEBT_VEBT > complex,G: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups1794756597179926696omplex @ F @ S2 ) ) @ ( groups2240296850493347238T_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7635_sum__norm__le,axiom,
    ! [S2: set_real,F: real > complex,G: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7636_sum__norm__le,axiom,
    ! [S2: set_int,F: int > complex,G: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7637_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > complex,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7638_sum__norm__le,axiom,
    ! [S2: set_complex,F: complex > complex,G: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7639_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > real,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7640_sum__mono,axiom,
    ! [K6: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K6 ) @ ( groups2906978787729119204at_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7641_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ K6 ) @ ( groups136491112297645522BT_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7642_sum__mono,axiom,
    ! [K6: set_real,F: real > rat,G: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K6 ) @ ( groups1300246762558778688al_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7643_sum__mono,axiom,
    ! [K6: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K6 ) @ ( groups3906332499630173760nt_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7644_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ K6 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ K6 ) @ ( groups771621172384141258BT_nat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7645_sum__mono,axiom,
    ! [K6: set_real,F: real > nat,G: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K6 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K6 ) @ ( groups1935376822645274424al_nat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7646_sum__mono,axiom,
    ! [K6: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K6 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K6 ) @ ( groups4541462559716669496nt_nat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7647_sum__mono,axiom,
    ! [K6: set_nat,F: nat > int,G: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K6 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K6 ) @ ( groups3539618377306564664at_int @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7648_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ K6 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups769130701875090982BT_int @ F @ K6 ) @ ( groups769130701875090982BT_int @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7649_sum__mono,axiom,
    ! [K6: set_real,F: real > int,G: real > int] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K6 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K6 ) @ ( groups1932886352136224148al_int @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_7650_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7651_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7652_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7653_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7654_sum__distrib__left,axiom,
    ! [R3: nat,F: nat > nat,A2: set_nat] :
      ( ( times_times_nat @ R3 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ R3 @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7655_sum__distrib__left,axiom,
    ! [R3: complex,F: complex > complex,A2: set_complex] :
      ( ( times_times_complex @ R3 @ ( groups7754918857620584856omplex @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( times_times_complex @ R3 @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7656_sum__distrib__left,axiom,
    ! [R3: int,F: int > int,A2: set_int] :
      ( ( times_times_int @ R3 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ R3 @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7657_sum__distrib__left,axiom,
    ! [R3: real,F: nat > real,A2: set_nat] :
      ( ( times_times_real @ R3 @ ( groups6591440286371151544t_real @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( times_times_real @ R3 @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7658_sum__distrib__right,axiom,
    ! [F: nat > nat,A2: set_nat,R3: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R3 )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7659_sum__distrib__right,axiom,
    ! [F: complex > complex,A2: set_complex,R3: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R3 )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( times_times_complex @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7660_sum__distrib__right,axiom,
    ! [F: int > int,A2: set_int,R3: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R3 )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7661_sum__distrib__right,axiom,
    ! [F: nat > real,A2: set_nat,R3: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R3 )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7662_sum__product,axiom,
    ! [F: nat > nat,A2: set_nat,G: nat > nat,B5: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_7663_sum__product,axiom,
    ! [F: complex > complex,A2: set_complex,G: complex > complex,B5: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I2: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_7664_sum__product,axiom,
    ! [F: int > int,A2: set_int,G: int > int,B5: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I2: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_7665_sum__product,axiom,
    ! [F: nat > real,A2: set_nat,G: nat > real,B5: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_7666_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
        @ A2 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_7667_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
        @ A2 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_7668_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
        @ A2 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_7669_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R3: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R3 )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( divide1717551699836669952omplex @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_7670_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R3: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R3 )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_7671_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_nat,G: vEBT_VEBT > nat > nat,R2: vEBT_VEBT > nat > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups771621172384141258BT_nat
            @ ^ [X: vEBT_VEBT] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups771621172384141258BT_nat
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7672_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B5: set_nat,G: real > nat > nat,R2: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X: real] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7673_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B5: set_nat,G: int > nat > nat,R2: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X: int] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X: int] : ( G @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7674_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B5: set_nat,G: complex > nat > nat,R2: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X: complex] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X: complex] : ( G @ X @ Y )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7675_sum_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B5: set_nat,G: code_integer > nat > nat,R2: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X: code_integer] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups7237345082560585321er_nat
                @ ^ [X: code_integer] : ( G @ X @ Y )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7676_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_complex,G: vEBT_VEBT > complex > complex,R2: vEBT_VEBT > complex > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups1794756597179926696omplex
            @ ^ [X: vEBT_VEBT] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups1794756597179926696omplex
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7677_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B5: set_complex,G: real > complex > complex,R2: real > complex > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups5754745047067104278omplex
            @ ^ [X: real] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7678_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B5: set_complex,G: nat > complex > complex,R2: nat > complex > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X: nat] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X: nat] : ( G @ X @ Y )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7679_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B5: set_complex,G: int > complex > complex,R2: int > complex > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X: int] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X: int] : ( G @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7680_sum_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B5: set_complex,G: code_integer > complex > complex,R2: code_integer > complex > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups8024822376189712711omplex
            @ ^ [X: code_integer] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups8024822376189712711omplex
                @ ^ [X: code_integer] : ( G @ X @ Y )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7681_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I2: nat] : ( modulo_modulo_nat @ ( F @ I2 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7682_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I2: int] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7683_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7684_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7685_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7686_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7687_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7688_sum__nonpos,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7689_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7690_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7691_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7692_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7693_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7694_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7695_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7696_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7697_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7698_sum__nonneg,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7699_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7700_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7701_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7702_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7703_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 ) )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7704_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7705_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7706_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7707_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7708_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 ) )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_Code_integer @ I @ I5 )
         => ( ( finite6017078050557962740nteger @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7709_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 ) )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7710_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7711_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7712_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7713_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7714_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7715_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > uint32,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups8325533452322294502uint32 @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups8325533452322294502uint32
          @ ^ [X: vEBT_VEBT] : ( if_uint32 @ ( P @ X ) @ ( G @ X ) @ zero_zero_uint32 )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7716_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > uint32,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5944083974425963860uint32 @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups5944083974425963860uint32
          @ ^ [X: real] : ( if_uint32 @ ( P @ X ) @ ( G @ X ) @ zero_zero_uint32 )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7717_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > uint32,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups833757482993574392uint32 @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups833757482993574392uint32
          @ ^ [X: nat] : ( if_uint32 @ ( P @ X ) @ ( G @ X ) @ zero_zero_uint32 )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7718_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > uint32,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups5712668689793887828uint32 @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups5712668689793887828uint32
          @ ^ [X: int] : ( if_uint32 @ ( P @ X ) @ ( G @ X ) @ zero_zero_uint32 )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7719_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > uint32,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups8736914816313324502uint32 @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups8736914816313324502uint32
          @ ^ [X: complex] : ( if_uint32 @ ( P @ X ) @ ( G @ X ) @ zero_zero_uint32 )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7720_sum_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: code_integer > uint32,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups8847630953604152069uint32 @ G
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ( ( member_Code_integer @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups8847630953604152069uint32
          @ ^ [X: code_integer] : ( if_uint32 @ ( P @ X ) @ ( G @ X ) @ zero_zero_uint32 )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7721_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2240296850493347238T_real @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups2240296850493347238T_real
          @ ^ [X: vEBT_VEBT] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7722_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7723_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7724_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7725_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
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7726_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
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7727_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
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7728_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
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7729_sum__le__included,axiom,
    ! [S3: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7730_sum__le__included,axiom,
    ! [S3: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ? [Xa2: complex] :
                    ( ( member_complex @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7731_sum__le__included,axiom,
    ! [S3: set_int,T: set_Code_integer,G: code_integer > real,I: code_integer > int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7732_sum__le__included,axiom,
    ! [S3: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7733_sum__le__included,axiom,
    ! [S3: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa2: complex] :
                    ( ( member_complex @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7734_sum__le__included,axiom,
    ! [S3: set_complex,T: set_Code_integer,G: code_integer > real,I: code_integer > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7735_sum__le__included,axiom,
    ! [S3: set_Code_integer,T: set_int,G: int > real,I: int > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7736_sum__le__included,axiom,
    ! [S3: set_Code_integer,T: set_complex,G: complex > real,I: complex > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ? [Xa2: complex] :
                    ( ( member_complex @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7737_sum__le__included,axiom,
    ! [S3: set_Code_integer,T: set_Code_integer,G: code_integer > real,I: code_integer > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S3 ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7738_sum__le__included,axiom,
    ! [S3: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7739_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7740_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7741_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7742_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7743_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: code_integer] :
                ( ( member_Code_integer @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7744_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7745_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7746_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7747_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7748_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7749_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A2 )
              & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7750_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
              & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7751_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ A2 )
              & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7752_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A2 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7753_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A2 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7754_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7755_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ A2 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7756_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A2 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7757_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7758_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ A2 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A2 ) @ ( groups7237345082560585321er_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7759_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S2: set_nat,H2: nat > uint32,G: nat > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X16: uint32,Y15: uint32,X24: uint32,Y24: uint32] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_uint32 @ X16 @ Y15 ) @ ( plus_plus_uint32 @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups833757482993574392uint32 @ H2 @ S2 ) @ ( groups833757482993574392uint32 @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7760_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S2: set_int,H2: int > uint32,G: int > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X16: uint32,Y15: uint32,X24: uint32,Y24: uint32] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_uint32 @ X16 @ Y15 ) @ ( plus_plus_uint32 @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5712668689793887828uint32 @ H2 @ S2 ) @ ( groups5712668689793887828uint32 @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7761_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S2: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X16: uint32,Y15: uint32,X24: uint32,Y24: uint32] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_uint32 @ X16 @ Y15 ) @ ( plus_plus_uint32 @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups8736914816313324502uint32 @ H2 @ S2 ) @ ( groups8736914816313324502uint32 @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7762_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S2: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X16: uint32,Y15: uint32,X24: uint32,Y24: uint32] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_uint32 @ X16 @ Y15 ) @ ( plus_plus_uint32 @ X24 @ Y24 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups8847630953604152069uint32 @ H2 @ S2 ) @ ( groups8847630953604152069uint32 @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7763_sum_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X24: real,Y24: real] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7764_sum_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X24: real,Y24: real] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7765_sum_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X24: real,Y24: real] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X24 @ Y24 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups1270011288395367621r_real @ H2 @ S2 ) @ ( groups1270011288395367621r_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7766_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X24: rat,Y24: rat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7767_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X24: rat,Y24: rat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7768_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X24: rat,Y24: rat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7769_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7770_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7771_sum__strict__mono,axiom,
    ! [A2: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7772_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7773_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7774_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7775_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7776_sum__strict__mono,axiom,
    ! [A2: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7777_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7778_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7779_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( groups2240296850493347238T_real @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( plus_plus_real @ ( G @ X2 ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7780_sum_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X2 @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X2 @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X2 @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X2 @ A2 ) )
            = ( plus_plus_real @ ( G @ X2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7781_sum_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X2 @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X2 @ A2 ) )
            = ( groups8778361861064173332t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X2 @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X2 @ A2 ) )
            = ( plus_plus_real @ ( G @ X2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7782_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X2: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X2 @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X2 @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X2 @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X2 @ A2 ) )
            = ( plus_plus_real @ ( G @ X2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7783_sum_Oinsert__if,axiom,
    ! [A2: set_Code_integer,X2: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ X2 @ A2 )
         => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X2 @ A2 ) )
            = ( groups1270011288395367621r_real @ G @ A2 ) ) )
        & ( ~ ( member_Code_integer @ X2 @ A2 )
         => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X2 @ A2 ) )
            = ( plus_plus_real @ ( G @ X2 ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7784_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( groups136491112297645522BT_rat @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X2 ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7785_sum_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X2 @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X2 @ A2 ) )
            = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X2 @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X2 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7786_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X2: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X2 @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X2 @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X2 @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X2 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7787_sum_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X2 @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X2 @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X2 @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X2 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7788_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X2: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X2 @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X2 @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X2 @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X2 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7789_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: vEBT_VEBT > vEBT_VEBT,J: vEBT_VEBT > vEBT_VEBT,T5: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S2 )
                        = ( groups8325533452322294502uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7790_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_real,S2: set_VEBT_VEBT,I: real > vEBT_VEBT,J: vEBT_VEBT > real,T5: set_real,G: vEBT_VEBT > uint32,H2: real > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_real @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T5 @ T7 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S2 )
                        = ( groups5944083974425963860uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7791_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_VEBT_VEBT,S2: set_real,I: vEBT_VEBT > real,J: real > vEBT_VEBT,T5: set_VEBT_VEBT,G: real > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S2 )
                        = ( groups8325533452322294502uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7792_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_real,S2: set_real,I: real > real,J: real > real,T5: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_real @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T5 @ T7 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S2 )
                        = ( groups5944083974425963860uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7793_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_int,S2: set_VEBT_VEBT,I: int > vEBT_VEBT,J: vEBT_VEBT > int,T5: set_int,G: vEBT_VEBT > uint32,H2: int > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_int @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T7 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S2 )
                        = ( groups5712668689793887828uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7794_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_int,S2: set_real,I: int > real,J: real > int,T5: set_int,G: real > uint32,H2: int > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_int @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T7 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S2 )
                        = ( groups5712668689793887828uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7795_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_complex,S2: set_VEBT_VEBT,I: complex > vEBT_VEBT,J: vEBT_VEBT > complex,T5: set_complex,G: vEBT_VEBT > uint32,H2: complex > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite3207457112153483333omplex @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T7 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S2 )
                        = ( groups8736914816313324502uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7796_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_complex,S2: set_real,I: complex > real,J: real > complex,T5: set_complex,G: real > uint32,H2: complex > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite3207457112153483333omplex @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T7 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S2 )
                        = ( groups8736914816313324502uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7797_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_Code_integer,S2: set_VEBT_VEBT,I: code_integer > vEBT_VEBT,J: vEBT_VEBT > code_integer,T5: set_Code_integer,G: vEBT_VEBT > uint32,H2: code_integer > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite6017078050557962740nteger @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T5 @ T7 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S2 )
                        = ( groups8847630953604152069uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7798_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_Code_integer,S2: set_real,I: code_integer > real,J: real > code_integer,T5: set_Code_integer,G: real > uint32,H2: code_integer > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite6017078050557962740nteger @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T5 @ T7 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S2 )
                        = ( groups8847630953604152069uint32 @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7799_sum__nonneg__0,axiom,
    ! [S3: set_VEBT_VEBT,F: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_VEBT_VEBT @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7800_sum__nonneg__0,axiom,
    ! [S3: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_real @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7801_sum__nonneg__0,axiom,
    ! [S3: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_int @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7802_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_complex @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7803_sum__nonneg__0,axiom,
    ! [S3: set_Code_integer,F: code_integer > real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_Code_integer @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7804_sum__nonneg__0,axiom,
    ! [S3: set_VEBT_VEBT,F: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_VEBT_VEBT @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7805_sum__nonneg__0,axiom,
    ! [S3: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_real @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7806_sum__nonneg__0,axiom,
    ! [S3: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7807_sum__nonneg__0,axiom,
    ! [S3: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_int @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7808_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7809_sum__nonneg__leq__bound,axiom,
    ! [S3: set_VEBT_VEBT,F: vEBT_VEBT > real,B5: real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S3 )
            = B5 )
         => ( ( member_VEBT_VEBT @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7810_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: real > real,B5: real,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S3 )
            = B5 )
         => ( ( member_real @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7811_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: int > real,B5: real,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S3 )
            = B5 )
         => ( ( member_int @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7812_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: complex > real,B5: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S3 )
            = B5 )
         => ( ( member_complex @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7813_sum__nonneg__leq__bound,axiom,
    ! [S3: set_Code_integer,F: code_integer > real,B5: real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S3 )
            = B5 )
         => ( ( member_Code_integer @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7814_sum__nonneg__leq__bound,axiom,
    ! [S3: set_VEBT_VEBT,F: vEBT_VEBT > rat,B5: rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S3 )
            = B5 )
         => ( ( member_VEBT_VEBT @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7815_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: real > rat,B5: rat,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
            = B5 )
         => ( ( member_real @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7816_sum__nonneg__leq__bound,axiom,
    ! [S3: set_nat,F: nat > rat,B5: rat,I: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
            = B5 )
         => ( ( member_nat @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7817_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: int > rat,B5: rat,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
            = B5 )
         => ( ( member_int @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7818_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: complex > rat,B5: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
            = B5 )
         => ( ( member_complex @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7819_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > uint32] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups5712668689793887828uint32 @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_uint32 ) ) ) )
        = ( groups5712668689793887828uint32 @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7820_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups8736914816313324502uint32 @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_uint32 ) ) ) )
        = ( groups8736914816313324502uint32 @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7821_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups8847630953604152069uint32 @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X: code_integer] :
                  ( ( G @ X )
                  = zero_zero_uint32 ) ) ) )
        = ( groups8847630953604152069uint32 @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7822_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7823_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7824_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups1270011288395367621r_real @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X: code_integer] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups1270011288395367621r_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7825_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7826_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7827_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X: code_integer] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7828_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7829_sum__power__add,axiom,
    ! [X2: uint32,M: nat,I5: set_nat] :
      ( ( groups833757482993574392uint32
        @ ^ [I2: nat] : ( power_power_uint32 @ X2 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_uint32 @ ( power_power_uint32 @ X2 @ M ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7830_sum__power__add,axiom,
    ! [X2: word_N3645301735248828278l_num1,M: nat,I5: set_nat] :
      ( ( groups2996710295995929986l_num1
        @ ^ [I2: nat] : ( power_2184487114949457152l_num1 @ X2 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ M ) @ ( groups2996710295995929986l_num1 @ ( power_2184487114949457152l_num1 @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7831_sum__power__add,axiom,
    ! [X2: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I2: nat] : ( power_power_rat @ X2 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7832_sum__power__add,axiom,
    ! [X2: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7833_sum__power__add,axiom,
    ! [X2: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7834_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7835_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7836_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 ) )
         => ( ! [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7837_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 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7838_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 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7839_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 ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7840_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 ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7841_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 ) )
         => ( ! [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7842_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 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7843_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 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7844_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 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7845_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 ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7846_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7847_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7848_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7849_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7850_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7851_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7852_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7853_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups6602215022474089585er_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7854_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7855_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7856_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8325533452322294502uint32 @ G @ A2 )
                  = ( groups8325533452322294502uint32 @ H2 @ B5 ) )
                = ( ( groups8325533452322294502uint32 @ G @ C4 )
                  = ( groups8325533452322294502uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7857_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups5944083974425963860uint32 @ G @ A2 )
                  = ( groups5944083974425963860uint32 @ H2 @ B5 ) )
                = ( ( groups5944083974425963860uint32 @ G @ C4 )
                  = ( groups5944083974425963860uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7858_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8736914816313324502uint32 @ G @ A2 )
                  = ( groups8736914816313324502uint32 @ H2 @ B5 ) )
                = ( ( groups8736914816313324502uint32 @ G @ C4 )
                  = ( groups8736914816313324502uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7859_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8847630953604152069uint32 @ G @ A2 )
                  = ( groups8847630953604152069uint32 @ H2 @ B5 ) )
                = ( ( groups8847630953604152069uint32 @ G @ C4 )
                  = ( groups8847630953604152069uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7860_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ A2 )
                  = ( groups2240296850493347238T_real @ H2 @ B5 ) )
                = ( ( groups2240296850493347238T_real @ G @ C4 )
                  = ( groups2240296850493347238T_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7861_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B5 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7862_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B5 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7863_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ A2 )
                  = ( groups1270011288395367621r_real @ H2 @ B5 ) )
                = ( ( groups1270011288395367621r_real @ G @ C4 )
                  = ( groups1270011288395367621r_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7864_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ A2 )
                  = ( groups136491112297645522BT_rat @ H2 @ B5 ) )
                = ( ( groups136491112297645522BT_rat @ G @ C4 )
                  = ( groups136491112297645522BT_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7865_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B5 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7866_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8325533452322294502uint32 @ G @ C4 )
                  = ( groups8325533452322294502uint32 @ H2 @ C4 ) )
               => ( ( groups8325533452322294502uint32 @ G @ A2 )
                  = ( groups8325533452322294502uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7867_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups5944083974425963860uint32 @ G @ C4 )
                  = ( groups5944083974425963860uint32 @ H2 @ C4 ) )
               => ( ( groups5944083974425963860uint32 @ G @ A2 )
                  = ( groups5944083974425963860uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7868_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8736914816313324502uint32 @ G @ C4 )
                  = ( groups8736914816313324502uint32 @ H2 @ C4 ) )
               => ( ( groups8736914816313324502uint32 @ G @ A2 )
                  = ( groups8736914816313324502uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7869_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8847630953604152069uint32 @ G @ C4 )
                  = ( groups8847630953604152069uint32 @ H2 @ C4 ) )
               => ( ( groups8847630953604152069uint32 @ G @ A2 )
                  = ( groups8847630953604152069uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7870_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ C4 )
                  = ( groups2240296850493347238T_real @ H2 @ C4 ) )
               => ( ( groups2240296850493347238T_real @ G @ A2 )
                  = ( groups2240296850493347238T_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7871_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7872_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7873_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ C4 )
                  = ( groups1270011288395367621r_real @ H2 @ C4 ) )
               => ( ( groups1270011288395367621r_real @ G @ A2 )
                  = ( groups1270011288395367621r_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7874_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ C4 )
                  = ( groups136491112297645522BT_rat @ H2 @ C4 ) )
               => ( ( groups136491112297645522BT_rat @ G @ A2 )
                  = ( groups136491112297645522BT_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7875_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7876_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8736914816313324502uint32 @ G @ S2 )
            = ( groups8736914816313324502uint32 @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7877_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8847630953604152069uint32 @ G @ S2 )
            = ( groups8847630953604152069uint32 @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7878_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S2 )
            = ( groups5808333547571424918x_real @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7879_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ S2 )
            = ( groups1270011288395367621r_real @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7880_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S2 )
            = ( groups5058264527183730370ex_rat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7881_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ S2 )
            = ( groups6602215022474089585er_rat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7882_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S2 )
            = ( groups5693394587270226106ex_nat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7883_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ S2 )
            = ( groups7237345082560585321er_nat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7884_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S2 )
            = ( groups5690904116761175830ex_int @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7885_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ S2 )
            = ( groups7234854612051535045er_int @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7886_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8736914816313324502uint32 @ G @ T5 )
            = ( groups8736914816313324502uint32 @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7887_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8847630953604152069uint32 @ G @ T5 )
            = ( groups8847630953604152069uint32 @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7888_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T5 )
            = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7889_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ T5 )
            = ( groups1270011288395367621r_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7890_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T5 )
            = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7891_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ T5 )
            = ( groups6602215022474089585er_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7892_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T5 )
            = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7893_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ T5 )
            = ( groups7237345082560585321er_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7894_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T5 )
            = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7895_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ T5 )
            = ( groups7234854612051535045er_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7896_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: vEBT_VEBT > uint32,G: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T5 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8325533452322294502uint32 @ G @ S2 )
              = ( groups8325533452322294502uint32 @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7897_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S2: set_real,H2: real > uint32,G: real > uint32] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S2 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5944083974425963860uint32 @ G @ S2 )
              = ( groups5944083974425963860uint32 @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7898_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S2: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8736914816313324502uint32 @ G @ S2 )
              = ( groups8736914816313324502uint32 @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7899_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8847630953604152069uint32 @ G @ S2 )
              = ( groups8847630953604152069uint32 @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7900_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T5 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ S2 )
              = ( groups2240296850493347238T_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7901_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S2: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S2 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S2 )
              = ( groups8097168146408367636l_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7902_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S2 )
              = ( groups5808333547571424918x_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7903_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ S2 )
              = ( groups1270011288395367621r_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7904_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T5 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ S2 )
              = ( groups136491112297645522BT_rat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7905_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S2: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S2 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S2 )
              = ( groups1300246762558778688al_rat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7906_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T5 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8325533452322294502uint32 @ G @ T5 )
              = ( groups8325533452322294502uint32 @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7907_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S2: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S2 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5944083974425963860uint32 @ G @ T5 )
              = ( groups5944083974425963860uint32 @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7908_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8736914816313324502uint32 @ G @ T5 )
              = ( groups8736914816313324502uint32 @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7909_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8847630953604152069uint32 @ G @ T5 )
              = ( groups8847630953604152069uint32 @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7910_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T5 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ T5 )
              = ( groups2240296850493347238T_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7911_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S2 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T5 )
              = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7912_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T5 )
              = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7913_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ T5 )
              = ( groups1270011288395367621r_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7914_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T5 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T5 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ T5 )
              = ( groups136491112297645522BT_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7915_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S2 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T5 )
              = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7916_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7917_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > real] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups1270011288395367621r_real @ G @ A2 )
          = ( plus_plus_real @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups1270011288395367621r_real @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7918_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7919_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups6602215022474089585er_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups6602215022474089585er_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7920_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7921_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups7237345082560585321er_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups7237345082560585321er_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7922_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7923_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups7234854612051535045er_int @ G @ A2 )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups7234854612051535045er_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7924_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7925_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7926_sum__diff,axiom,
    ! [A2: set_complex,B5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7927_sum__diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7928_sum__diff,axiom,
    ! [A2: set_complex,B5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7929_sum__diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7930_sum__diff,axiom,
    ! [A2: set_complex,B5: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7931_sum__diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A2 ) @ ( groups7234854612051535045er_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7932_sum__diff,axiom,
    ! [A2: set_nat,B5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B5 @ A2 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7933_sum__diff,axiom,
    ! [A2: set_nat,B5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B5 @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7934_sum__diff,axiom,
    ! [A2: set_int,B5: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7935_sum__diff,axiom,
    ! [A2: set_int,B5: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_7936_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_7937_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > uint32,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_uint32 )
     => ( ( groups833757482993574392uint32 @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups833757482993574392uint32 @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_7938_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_7939_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_7940_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_7941_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_7942_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_7943_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_7944_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_7945_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_7946_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_7947_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_7948_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_7949_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_7950_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_7951_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_7952_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_7953_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_7954_dbl__inc__def,axiom,
    ( neg_nu8115118780965096967l_num1
    = ( ^ [X: word_N3645301735248828278l_num1] : ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ X @ X ) @ one_on7727431528512463931l_num1 ) ) ) ).

% dbl_inc_def
thf(fact_7955_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_7956_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_7957_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_7958_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
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7959_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
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7960_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
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7961_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
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7962_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7963_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7964_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( minus_minus_real @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7965_sum__atLeastAtMost__code,axiom,
    ! [F: nat > uint32,A: nat,B: nat] :
      ( ( groups833757482993574392uint32 @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo8366116489143299838uint32
        @ ^ [A4: nat] : ( plus_plus_uint32 @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_uint32 ) ) ).

% sum_atLeastAtMost_code
thf(fact_7966_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7967_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_7968_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7969_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_7970_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7971_sum__mono2,axiom,
    ! [B5: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [B2: real] :
              ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7972_sum__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [B2: complex] :
              ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7973_sum__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7974_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7975_sum__mono2,axiom,
    ! [B5: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [B2: real] :
              ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7976_sum__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [B2: complex] :
              ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7977_sum__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7978_sum__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ! [B2: nat] :
              ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7979_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
         => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( groups771621172384141258BT_nat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_7980_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,X2: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
        = ( plus_plus_real @ ( G @ X2 ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7981_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X2: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X2 @ A2 ) )
        = ( plus_plus_real @ ( G @ X2 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7982_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > real,X2: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X2 @ A2 ) )
        = ( plus_plus_real @ ( G @ X2 ) @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7983_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > rat,X2: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
        = ( plus_plus_rat @ ( G @ X2 ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7984_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X2: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X2 @ A2 ) )
        = ( plus_plus_rat @ ( G @ X2 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7985_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat,X2: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X2 @ A2 ) )
        = ( plus_plus_rat @ ( G @ X2 ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7986_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > nat,X2: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
        = ( plus_plus_nat @ ( G @ X2 ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7987_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X2: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X2 @ A2 ) )
        = ( plus_plus_nat @ ( G @ X2 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7988_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > nat,X2: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X2 @ A2 ) )
        = ( plus_plus_nat @ ( G @ X2 ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7989_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > int,X2: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups769130701875090982BT_int @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
        = ( plus_plus_int @ ( G @ X2 ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7990_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7991_sum_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X2 @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7992_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X2: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X2 @ A2 )
       => ( ( groups1270011288395367621r_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X2 ) @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7993_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7994_sum_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X2 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7995_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X2: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X2 @ A2 )
       => ( ( groups6602215022474089585er_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X2 ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7996_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups771621172384141258BT_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X2 ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7997_sum_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X2 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X2 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7998_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X2: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X2 @ A2 )
       => ( ( groups7237345082560585321er_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X2 ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7999_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups769130701875090982BT_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X2 ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8000_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups2240296850493347238T_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8001_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8002_sum__diff1,axiom,
    ! [A2: set_Code_integer,A: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ A @ A2 )
         => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Code_integer @ A @ A2 )
         => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( groups1270011288395367621r_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8003_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8004_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8005_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups136491112297645522BT_rat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups136491112297645522BT_rat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8006_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8007_sum__diff1,axiom,
    ! [A2: set_Code_integer,A: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ A @ A2 )
         => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Code_integer @ A @ A2 )
         => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( groups6602215022474089585er_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8008_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8009_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8010_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( 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 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8011_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( 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 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8012_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( 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 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8013_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( 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 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8014_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real,C: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8015_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K4: complex] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K4: complex] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8016_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > real,C: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K4: code_integer] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups1270011288395367621r_real @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K4: code_integer] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups1270011288395367621r_real @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8017_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat,C: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K4: vEBT_VEBT] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K4: vEBT_VEBT] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8018_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K4: complex] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K4: complex] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8019_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > rat,C: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K4: code_integer] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K4: code_integer] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8020_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > nat,C: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K4: vEBT_VEBT] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K4: vEBT_VEBT] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8021_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K4: complex] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K4: complex] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8022_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > nat,C: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K4: code_integer] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K4: code_integer] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8023_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > int,C: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K4: vEBT_VEBT] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K4: vEBT_VEBT] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8024_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8025_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8026_sum__strict__mono2,axiom,
    ! [B5: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8027_sum__strict__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8028_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,B: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8029_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8030_sum__strict__mono2,axiom,
    ! [B5: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8031_sum__strict__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8032_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,B: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8033_sum__strict__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,B: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8034_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( groups771621172384141258BT_nat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8035_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups2240296850493347238T_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8036_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8037_member__le__sum,axiom,
    ! [I: code_integer,A2: set_Code_integer,F: code_integer > real] :
      ( ( member_Code_integer @ I @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups1270011288395367621r_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8038_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > real] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8039_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > real] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8040_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8041_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8042_member__le__sum,axiom,
    ! [I: code_integer,A2: set_Code_integer,F: code_integer > rat] :
      ( ( member_Code_integer @ I @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups6602215022474089585er_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8043_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > rat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8044_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > rat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8045_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > uint32] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups833757482993574392uint32
            @ ^ [K4: nat] : ( minus_minus_uint32 @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_uint32 @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups833757482993574392uint32
            @ ^ [K4: nat] : ( minus_minus_uint32 @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_uint32 ) ) ) ).

% sum_natinterval_diff
thf(fact_8046_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K4: nat] : ( minus_minus_rat @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K4: nat] : ( minus_minus_rat @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_8047_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K4: nat] : ( minus_minus_int @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K4: nat] : ( minus_minus_int @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_8048_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K4: nat] : ( minus_minus_real @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K4: nat] : ( minus_minus_real @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_8049_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K4: nat] : ( minus_minus_rat @ ( F @ K4 ) @ ( F @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8050_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K4: nat] : ( minus_minus_int @ ( F @ K4 ) @ ( F @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8051_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K4: nat] : ( minus_minus_real @ ( F @ K4 ) @ ( F @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8052_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) @ one_one_uint32 )
      = ( groups833757482993574392uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8053_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) @ one_on7727431528512463931l_num1 )
      = ( groups2996710295995929986l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8054_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8055_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8056_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: uint32] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X2 ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_uint32 @ ( power_power_uint32 @ X2 @ M ) @ ( power_power_uint32 @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8057_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ X2 ) @ ( groups2996710295995929986l_num1 @ ( power_2184487114949457152l_num1 @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ M ) @ ( power_2184487114949457152l_num1 @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8058_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8059_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8060_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8061_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I2: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_8062_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_8063_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_8064_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_8065_VEBT__internal_Oheight_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A3: $o,B2: $o] :
          ( X2
         != ( vEBT_Leaf @ A3 @ B2 ) )
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).

% VEBT_internal.height.cases
thf(fact_8066_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_8067_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_8068_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
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_8069_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_8070_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( groups2996710295995929986l_num1 @ semiri8819519690708144855l_num1 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7065122842183080059l_num1 @ ( semiri8819519690708144855l_num1 @ N2 ) @ ( plus_p361126936061061375l_num1 @ ( semiri8819519690708144855l_num1 @ N2 ) @ one_on7727431528512463931l_num1 ) ) ) ).

% double_gauss_sum
thf(fact_8071_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_8072_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_8073_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum
thf(fact_8074_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_8075_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_8076_double__arith__series,axiom,
    ! [A: word_N3645301735248828278l_num1,D: word_N3645301735248828278l_num1,N2: nat] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) )
        @ ( groups2996710295995929986l_num1
          @ ^ [I2: nat] : ( plus_p361126936061061375l_num1 @ A @ ( times_7065122842183080059l_num1 @ ( semiri8819519690708144855l_num1 @ I2 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ ( semiri8819519690708144855l_num1 @ N2 ) @ one_on7727431528512463931l_num1 ) @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A ) @ ( times_7065122842183080059l_num1 @ ( semiri8819519690708144855l_num1 @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8077_double__arith__series,axiom,
    ! [A: rat,D: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I2 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8078_double__arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8079_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
        @ ( groups7501900531339628137nteger
          @ ^ [I2: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I2 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8080_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8081_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I2 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8082_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_8083_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_8084_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( groups2996710295995929986l_num1 @ semiri8819519690708144855l_num1 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_7065122842183080059l_num1 @ ( semiri8819519690708144855l_num1 @ N2 ) @ ( plus_p361126936061061375l_num1 @ ( semiri8819519690708144855l_num1 @ N2 ) @ one_on7727431528512463931l_num1 ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8085_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8086_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8087_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8088_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8089_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8090_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_8091_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_8092_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_8093_arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8094_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I2: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I2 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8095_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8096_sum__gp__offset,axiom,
    ! [X2: rat,M: nat,N2: nat] :
      ( ( ( X2 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X2 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8097_sum__gp__offset,axiom,
    ! [X2: real,M: nat,N2: nat] :
      ( ( ( X2 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X2 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8098_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8099_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8100_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8101_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_8102_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_8103_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_8104_sums__zero,axiom,
    ( sums_real
    @ ^ [N: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_8105_sums__zero,axiom,
    ( sums_nat
    @ ^ [N: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_8106_sums__zero,axiom,
    ( sums_int
    @ ^ [N: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_8107_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S2: real,A2: set_nat,S7: real,F: nat > real] :
      ( ( sums_real @ G @ S2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S7
            = ( plus_plus_real @ S2
              @ ( groups6591440286371151544t_real
                @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
                @ A2 ) ) )
         => ( sums_real
            @ ^ [N: nat] : ( if_real @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
            @ S7 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_8108_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8109_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8110_int__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% int_sum
thf(fact_8111_int__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% int_sum
thf(fact_8112_sum__subtractf__nat,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > nat,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups771621172384141258BT_nat
          @ ^ [X: vEBT_VEBT] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( groups771621172384141258BT_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8113_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8114_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8115_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8116_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N2 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_8117_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: int] :
                  ( ( member_int @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8118_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8119_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( groups7237345082560585321er_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: code_integer] :
              ( ( member_Code_integer @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: code_integer] :
                  ( ( member_Code_integer @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8120_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8121_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: int] :
                  ( ( member_int @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8122_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8123_sum__eq__1__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( groups7237345082560585321er_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: code_integer] :
              ( ( member_Code_integer @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: code_integer] :
                  ( ( member_Code_integer @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8124_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8125_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8126_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8127_sum__diff__nat,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8128_sum__diff__nat,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups7237345082560585321er_nat @ F @ A2 ) @ ( groups7237345082560585321er_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8129_sum__diff__nat,axiom,
    ! [B5: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8130_sum__diff__nat,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8131_sum__diff1__nat,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ( member_VEBT_VEBT @ A @ A2 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( minus_minus_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ A @ A2 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( groups771621172384141258BT_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8132_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8133_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8134_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8135_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S3: real,T: real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( sums_real @ F @ S3 )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S3 @ T ) ) ) ) ).

% sums_le
thf(fact_8136_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S3: nat,T: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( sums_nat @ F @ S3 )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S3 @ T ) ) ) ) ).

% sums_le
thf(fact_8137_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S3: int,T: int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( sums_int @ F @ S3 )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S3 @ T ) ) ) ) ).

% sums_le
thf(fact_8138_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R: nat] : ( if_real @ ( R = I ) @ ( F @ R ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8139_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R: nat] : ( if_nat @ ( R = I ) @ ( F @ R ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8140_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R: nat] : ( if_int @ ( R = I ) @ ( F @ R ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8141_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8142_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8143_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8144_sums__mult2,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
        @ ( times_times_real @ A @ C ) ) ) ).

% sums_mult2
thf(fact_8145_sums__mult,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
        @ ( times_times_real @ C @ A ) ) ) ).

% sums_mult
thf(fact_8146_sums__diff,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
          @ ( minus_minus_real @ A @ B ) ) ) ) ).

% sums_diff
thf(fact_8147_sums__divide,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C )
        @ ( divide_divide_real @ A @ C ) ) ) ).

% sums_divide
thf(fact_8148_sums__sum,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > nat > real,X2: vEBT_VEBT > real] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ I5 )
         => ( sums_real @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
     => ( sums_real
        @ ^ [N: nat] :
            ( groups2240296850493347238T_real
            @ ^ [I2: vEBT_VEBT] : ( F @ I2 @ N )
            @ I5 )
        @ ( groups2240296850493347238T_real @ X2 @ I5 ) ) ) ).

% sums_sum
thf(fact_8149_sums__sum,axiom,
    ! [I5: set_real,F: real > nat > real,X2: real > real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( sums_real @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
     => ( sums_real
        @ ^ [N: nat] :
            ( groups8097168146408367636l_real
            @ ^ [I2: real] : ( F @ I2 @ N )
            @ I5 )
        @ ( groups8097168146408367636l_real @ X2 @ I5 ) ) ) ).

% sums_sum
thf(fact_8150_sums__sum,axiom,
    ! [I5: set_int,F: int > nat > real,X2: int > real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( sums_real @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
     => ( sums_real
        @ ^ [N: nat] :
            ( groups8778361861064173332t_real
            @ ^ [I2: int] : ( F @ I2 @ N )
            @ I5 )
        @ ( groups8778361861064173332t_real @ X2 @ I5 ) ) ) ).

% sums_sum
thf(fact_8151_sums__sum,axiom,
    ! [I5: set_nat,F: nat > nat > nat,X2: nat > nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( sums_nat @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
     => ( sums_nat
        @ ^ [N: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I2: nat] : ( F @ I2 @ N )
            @ I5 )
        @ ( groups3542108847815614940at_nat @ X2 @ I5 ) ) ) ).

% sums_sum
thf(fact_8152_sums__sum,axiom,
    ! [I5: set_complex,F: complex > nat > complex,X2: complex > complex] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( sums_complex @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
     => ( sums_complex
        @ ^ [N: nat] :
            ( groups7754918857620584856omplex
            @ ^ [I2: complex] : ( F @ I2 @ N )
            @ I5 )
        @ ( groups7754918857620584856omplex @ X2 @ I5 ) ) ) ).

% sums_sum
thf(fact_8153_sums__sum,axiom,
    ! [I5: set_int,F: int > nat > int,X2: int > int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( sums_int @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
     => ( sums_int
        @ ^ [N: nat] :
            ( groups4538972089207619220nt_int
            @ ^ [I2: int] : ( F @ I2 @ N )
            @ I5 )
        @ ( groups4538972089207619220nt_int @ X2 @ I5 ) ) ) ).

% sums_sum
thf(fact_8154_sums__sum,axiom,
    ! [I5: set_nat,F: nat > nat > real,X2: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( sums_real @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
     => ( sums_real
        @ ^ [N: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( F @ I2 @ N )
            @ I5 )
        @ ( groups6591440286371151544t_real @ X2 @ I5 ) ) ) ).

% sums_sum
thf(fact_8155_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_8156_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_8157_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_8158_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_8159_sums__Suc__imp,axiom,
    ! [F: nat > real,S3: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) )
          @ S3 )
       => ( sums_real @ F @ S3 ) ) ) ).

% sums_Suc_imp
thf(fact_8160_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8161_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8162_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8163_sums__Suc__iff,axiom,
    ! [F: nat > real,S3: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ S3 )
      = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_8164_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S3: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ( F @ I3 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
          @ S3 )
        = ( sums_real @ F @ S3 ) ) ) ).

% sums_zero_iff_shift
thf(fact_8165_sums__finite,axiom,
    ! [N3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_int ) )
       => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N3 ) ) ) ) ).

% sums_finite
thf(fact_8166_sums__finite,axiom,
    ! [N3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_nat ) )
       => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N3 ) ) ) ) ).

% sums_finite
thf(fact_8167_sums__finite,axiom,
    ! [N3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_real ) )
       => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N3 ) ) ) ) ).

% sums_finite
thf(fact_8168_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_int
        @ ^ [R: nat] : ( if_int @ ( P @ R ) @ ( F @ R ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8169_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_nat
        @ ^ [R: nat] : ( if_nat @ ( P @ R ) @ ( F @ R ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8170_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_real
        @ ^ [R: nat] : ( if_real @ ( P @ R ) @ ( F @ R ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8171_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_int
        @ ^ [R: nat] : ( if_int @ ( member_nat @ R @ A2 ) @ ( F @ R ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8172_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_nat
        @ ^ [R: nat] : ( if_nat @ ( member_nat @ R @ A2 ) @ ( F @ R ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8173_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_real
        @ ^ [R: nat] : ( if_real @ ( member_nat @ R @ A2 ) @ ( F @ R ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8174_lemma__termdiff2,axiom,
    ! [H2: rat,Z: rat,N2: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N2 ) @ ( power_power_rat @ Z @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q5 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8175_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q5 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8176_TBOUND__fi_H__adm,axiom,
    ! [Foo: vEBT_VEBT > vEBT_VEBTi > nat > nat] :
      ( comple1745167176254620304_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi )
      @ ^ [Fi: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
        ! [X: vEBT_VEBT,Xa4: vEBT_VEBTi,Xb4: nat] : ( time_T5737551269749752165_VEBTi @ ( produc4062687810467144008_VEBTi @ ( produc2164094337957399884_VEBTi @ Fi ) @ X @ Xa4 @ Xb4 ) @ ( Foo @ X @ Xa4 @ Xb4 ) ) ) ).

% TBOUND_fi'_adm
thf(fact_8177_TBOUND__fi_H__adm,axiom,
    ! [Foo: vEBT_VEBT > vEBT_VEBTi > nat > nat] :
      ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [Fi: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: vEBT_VEBT,Xa4: vEBT_VEBTi,Xb4: nat] : ( time_T8353473612707095248on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Fi ) @ X @ Xa4 @ Xb4 ) @ ( Foo @ X @ Xa4 @ Xb4 ) ) ) ).

% TBOUND_fi'_adm
thf(fact_8178_TBOUND__fi_H__adm,axiom,
    ! [Foo: vEBT_VEBT > vEBT_VEBTi > nat > nat] :
      ( comple2654586775044187945Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [Fi: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
        ! [X: vEBT_VEBT,Xa4: vEBT_VEBTi,Xb4: nat] : ( time_TBOUND_o @ ( produc2663629013181010545Heap_o @ ( produc8381543706267210711Heap_o @ Fi ) @ X @ Xa4 @ Xb4 ) @ ( Foo @ X @ Xa4 @ Xb4 ) ) ) ).

% TBOUND_fi'_adm
thf(fact_8179_pochhammer__double,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8180_pochhammer__double,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8181_of__nat__code,axiom,
    ( semiri8819519690708144855l_num1
    = ( ^ [N: nat] :
          ( semiri2846968517960172219l_num1
          @ ^ [I2: word_N3645301735248828278l_num1] : ( plus_p361126936061061375l_num1 @ I2 @ one_on7727431528512463931l_num1 )
          @ N
          @ zero_z3563351764282998399l_num1 ) ) ) ).

% of_nat_code
thf(fact_8182_of__nat__code,axiom,
    ( semiri2565882477558803405uint32
    = ( ^ [N: nat] :
          ( semiri2064589214733661617uint32
          @ ^ [I2: uint32] : ( plus_plus_uint32 @ I2 @ one_one_uint32 )
          @ N
          @ zero_zero_uint32 ) ) ) ).

% of_nat_code
thf(fact_8183_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I2: rat] : ( plus_plus_rat @ I2 @ one_one_rat )
          @ N
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_8184_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I2: real] : ( plus_plus_real @ I2 @ one_one_real )
          @ N
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_8185_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_8186_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_8187_of__nat__code,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N: nat] :
          ( semiri4055485073559036834nteger
          @ ^ [I2: code_integer] : ( plus_p5714425477246183910nteger @ I2 @ one_one_Code_integer )
          @ N
          @ zero_z3403309356797280102nteger ) ) ) ).

% of_nat_code
thf(fact_8188_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8189_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_8190_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8191_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_8192_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8193_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_8194_lessThan__subset__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y2 ) )
      = ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_8195_lessThan__subset__iff,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y2 ) )
      = ( ord_less_eq_num @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_8196_lessThan__subset__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y2 ) )
      = ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_8197_lessThan__subset__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y2 ) )
      = ( ord_less_eq_int @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_8198_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_8199_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_8200_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_8201_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_8202_pochhammer__0,axiom,
    ! [A: word_N3645301735248828278l_num1] :
      ( ( comm_s6431939913906641691l_num1 @ A @ zero_zero_nat )
      = one_on7727431528512463931l_num1 ) ).

% pochhammer_0
thf(fact_8203_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_8204_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_8205_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_8206_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_8207_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8208_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8209_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8210_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8211_single__Diff__lessThan,axiom,
    ! [K: real] :
      ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
      = ( insert_real @ K @ bot_bot_set_real ) ) ).

% single_Diff_lessThan
thf(fact_8212_single__Diff__lessThan,axiom,
    ! [K: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
      = ( insert_nat @ K @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_8213_single__Diff__lessThan,axiom,
    ! [K: int] :
      ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
      = ( insert_int @ K @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_8214_floor__diff__of__int,axiom,
    ! [X2: real,Z: int] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_8215_floor__diff__of__int,axiom,
    ! [X2: rat,Z: int] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_8216_zero__le__floor,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% zero_le_floor
thf(fact_8217_zero__le__floor,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X2 ) ) ).

% zero_le_floor
thf(fact_8218_numeral__le__floor,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).

% numeral_le_floor
thf(fact_8219_numeral__le__floor,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).

% numeral_le_floor
thf(fact_8220_floor__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_8221_floor__less__zero,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ zero_zero_int )
      = ( ord_less_rat @ X2 @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_8222_floor__less__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_8223_floor__less__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_8224_zero__less__floor,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ one_one_real @ X2 ) ) ).

% zero_less_floor
thf(fact_8225_zero__less__floor,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ).

% zero_less_floor
thf(fact_8226_floor__le__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
      = ( ord_less_real @ X2 @ one_one_real ) ) ).

% floor_le_zero
thf(fact_8227_floor__le__zero,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ zero_zero_int )
      = ( ord_less_rat @ X2 @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_8228_one__le__floor,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ one_one_real @ X2 ) ) ).

% one_le_floor
thf(fact_8229_one__le__floor,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ).

% one_le_floor
thf(fact_8230_floor__less__one,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
      = ( ord_less_real @ X2 @ one_one_real ) ) ).

% floor_less_one
thf(fact_8231_floor__less__one,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
      = ( ord_less_rat @ X2 @ one_one_rat ) ) ).

% floor_less_one
thf(fact_8232_floor__diff__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_8233_floor__diff__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_8234_floor__numeral__power,axiom,
    ! [X2: num,N2: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ).

% floor_numeral_power
thf(fact_8235_floor__numeral__power,axiom,
    ! [X2: num,N2: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ).

% floor_numeral_power
thf(fact_8236_floor__diff__one,axiom,
    ! [X2: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_8237_floor__diff__one,axiom,
    ! [X2: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_8238_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_8239_numeral__less__floor,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).

% numeral_less_floor
thf(fact_8240_numeral__less__floor,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).

% numeral_less_floor
thf(fact_8241_floor__le__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X2 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_8242_floor__le__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_8243_one__less__floor,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ).

% one_less_floor
thf(fact_8244_one__less__floor,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) ) ).

% one_less_floor
thf(fact_8245_floor__le__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
      = ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8246_floor__le__one,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
      = ( ord_less_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8247_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_8248_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8249_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8250_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8251_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8252_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8253_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,D: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite5795047828879050333T_VEBT @ B5 )
       => ( finite5138865860212152956T_VEBT
          @ ( collec6585197662114348218T_VEBT
            @ ^ [F4: vEBT_VEBT > vEBT_VEBT] :
              ! [X: vEBT_VEBT] :
                ( ( ( member_VEBT_VEBT @ X @ A2 )
                 => ( member_VEBT_VEBT @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_VEBT_VEBT @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8254_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_real,D: real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_real @ B5 )
       => ( finite1104210157756803322T_real
          @ ( collec3757161770927152700T_real
            @ ^ [F4: vEBT_VEBT > real] :
              ! [X: vEBT_VEBT] :
                ( ( ( member_VEBT_VEBT @ X @ A2 )
                 => ( member_real @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_VEBT_VEBT @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8255_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B5: set_VEBT_VEBT,D: vEBT_VEBT] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite5795047828879050333T_VEBT @ B5 )
       => ( finite8914422552584188136T_VEBT
          @ ( collec2344002128899761706T_VEBT
            @ ^ [F4: real > vEBT_VEBT] :
              ! [X: real] :
                ( ( ( member_real @ X @ A2 )
                 => ( member_VEBT_VEBT @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_real @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8256_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B5: set_real,D: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_real @ B5 )
       => ( finite2300945044744945038l_real
          @ ( collect_real_real
            @ ^ [F4: real > real] :
              ! [X: real] :
                ( ( ( member_real @ X @ A2 )
                 => ( member_real @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_real @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8257_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_nat,D: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( finite3061610839650506142BT_nat
          @ ( collec7439586953676655584BT_nat
            @ ^ [F4: vEBT_VEBT > nat] :
              ! [X: vEBT_VEBT] :
                ( ( ( member_VEBT_VEBT @ X @ A2 )
                 => ( member_nat @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_VEBT_VEBT @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8258_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B5: set_nat,D: nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( finite3878561044930982962al_nat
          @ ( collect_real_nat
            @ ^ [F4: real > nat] :
              ! [X: real] :
                ( ( ( member_real @ X @ A2 )
                 => ( member_nat @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_real @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8259_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_int,D: int] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( finite8107131856996085242BT_int
          @ ( collec3261735934167458876BT_int
            @ ^ [F4: vEBT_VEBT > int] :
              ! [X: vEBT_VEBT] :
                ( ( ( member_VEBT_VEBT @ X @ A2 )
                 => ( member_int @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_VEBT_VEBT @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8260_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B5: set_int,D: int] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( finite8924082062276562062al_int
          @ ( collect_real_int
            @ ^ [F4: real > int] :
              ! [X: real] :
                ( ( ( member_real @ X @ A2 )
                 => ( member_int @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_real @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8261_finite__set__of__finite__funs,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_complex,D: complex] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( finite5075905776163990908omplex
          @ ( collec4282787479422967742omplex
            @ ^ [F4: vEBT_VEBT > complex] :
              ! [X: vEBT_VEBT] :
                ( ( ( member_VEBT_VEBT @ X @ A2 )
                 => ( member_complex @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_VEBT_VEBT @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8262_finite__set__of__finite__funs,axiom,
    ! [A2: set_real,B5: set_complex,D: complex] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( finite5484300831722202128omplex
          @ ( collect_real_complex
            @ ^ [F4: real > complex] :
              ! [X: real] :
                ( ( ( member_real @ X @ A2 )
                 => ( member_complex @ ( F4 @ X ) @ B5 ) )
                & ( ~ ( member_real @ X @ A2 )
                 => ( ( F4 @ X )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_8263_of__int__floor__le,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) @ X2 ) ).

% of_int_floor_le
thf(fact_8264_of__int__floor__le,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) @ X2 ) ).

% of_int_floor_le
thf(fact_8265_floor__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y2 ) ) ) ).

% floor_mono
thf(fact_8266_floor__mono,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y2 ) ) ) ).

% floor_mono
thf(fact_8267_floor__less__cancel,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y2 ) )
     => ( ord_less_real @ X2 @ Y2 ) ) ).

% floor_less_cancel
thf(fact_8268_floor__less__cancel,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y2 ) )
     => ( ord_less_rat @ X2 @ Y2 ) ) ).

% floor_less_cancel
thf(fact_8269_floor__le__ceiling,axiom,
    ! [X2: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).

% floor_le_ceiling
thf(fact_8270_floor__le__ceiling,axiom,
    ! [X2: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ).

% floor_le_ceiling
thf(fact_8271_lessThan__strict__subset__iff,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
      = ( ord_less_real @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8272_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
      = ( ord_less_rat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8273_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8274_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8275_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8276_pochhammer__pos,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8277_pochhammer__pos,axiom,
    ! [X2: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8278_pochhammer__pos,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X2 )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8279_pochhammer__pos,axiom,
    ! [X2: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8280_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8281_pochhammer__eq__0__mono,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8282_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8283_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N2 )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8284_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8285_floor__le__round,axiom,
    ! [X2: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim8280529875227126926d_real @ X2 ) ) ).

% floor_le_round
thf(fact_8286_floor__le__round,axiom,
    ! [X2: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ X2 ) ) ).

% floor_le_round
thf(fact_8287_admissible__heap,axiom,
    ! [P: produc3881548065746020326Ti_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o] :
      ( comple6931689918642796574on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
          ( ( heap_T306965388786959644on_nat @ ( F4 @ X ) @ H @ H9 @ R @ N )
         => ( P @ X @ H @ H9 @ R @ N ) ) ) ).

% admissible_heap
thf(fact_8288_admissible__heap,axiom,
    ! [P: produc3881548065746020326Ti_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o > nat > $o] :
      ( comple6491863954676465222Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
        ! [X: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R: $o,N: nat] :
          ( ( heap_Time_effect_o @ ( F4 @ X ) @ H @ H9 @ R @ N )
         => ( P @ X @ H @ H9 @ R @ N ) ) ) ).

% admissible_heap
thf(fact_8289_admissible__heap,axiom,
    ! [P: produc3960855945107176009Ti_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > vEBT_VEBTi > nat > $o] :
      ( comple1745167176254620304_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
        ! [X: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat] :
          ( ( heap_T2071195472996403633_VEBTi @ ( F4 @ X ) @ H @ H9 @ R @ N )
         => ( P @ X @ H @ H9 @ R @ N ) ) ) ).

% admissible_heap
thf(fact_8290_admissible__heap,axiom,
    ! [P: produc3960855945107176009Ti_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o] :
      ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
          ( ( heap_T306965388786959644on_nat @ ( F4 @ X ) @ H @ H9 @ R @ N )
         => ( P @ X @ H @ H9 @ R @ N ) ) ) ).

% admissible_heap
thf(fact_8291_admissible__heap,axiom,
    ! [P: produc3960855945107176009Ti_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o > nat > $o] :
      ( comple2654586775044187945Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
        ! [X: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H9: heap_e7401611519738050253t_unit,R: $o,N: nat] :
          ( ( heap_Time_effect_o @ ( F4 @ X ) @ H @ H9 @ R @ N )
         => ( P @ X @ H @ H9 @ R @ N ) ) ) ).

% admissible_heap
thf(fact_8292_le__floor__iff,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).

% le_floor_iff
thf(fact_8293_le__floor__iff,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).

% le_floor_iff
thf(fact_8294_floor__less__iff,axiom,
    ! [X2: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ Z )
      = ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_8295_floor__less__iff,axiom,
    ! [X2: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z )
      = ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_8296_floor__add__int,axiom,
    ! [X2: real,Z: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ Z )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ) ).

% floor_add_int
thf(fact_8297_floor__add__int,axiom,
    ! [X2: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_8298_int__add__floor,axiom,
    ! [Z: int,X2: real] :
      ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X2 ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ) ).

% int_add_floor
thf(fact_8299_int__add__floor,axiom,
    ! [Z: int,X2: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ) ).

% int_add_floor
thf(fact_8300_le__floor__add,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y2 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ Y2 ) ) ) ).

% le_floor_add
thf(fact_8301_le__floor__add,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y2 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ Y2 ) ) ) ).

% le_floor_add
thf(fact_8302_floor__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( X2
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X2 @ N2 ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X2 ) @ N2 ) ) ) ).

% floor_power
thf(fact_8303_floor__power,axiom,
    ! [X2: rat,N2: nat] :
      ( ( X2
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X2 @ N2 ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X2 ) @ N2 ) ) ) ).

% floor_power
thf(fact_8304_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_8305_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_8306_pochhammer__nonneg,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8307_pochhammer__nonneg,axiom,
    ! [X2: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8308_pochhammer__nonneg,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8309_pochhammer__nonneg,axiom,
    ! [X2: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8310_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s6431939913906641691l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
          = one_on7727431528512463931l_num1 ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s6431939913906641691l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
          = zero_z3563351764282998399l_num1 ) ) ) ).

% pochhammer_0_left
thf(fact_8311_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s6516030829397196305uint32 @ zero_zero_uint32 @ N2 )
          = one_one_uint32 ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s6516030829397196305uint32 @ zero_zero_uint32 @ N2 )
          = zero_zero_uint32 ) ) ) ).

% pochhammer_0_left
thf(fact_8312_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_8313_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_8314_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_8315_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_8316_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8317_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8318_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N2: int] :
      ( ! [X3: int] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N2 ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8319_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8320_of__nat__floor,axiom,
    ! [R3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R3 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R3 ) ) ) @ R3 ) ) ).

% of_nat_floor
thf(fact_8321_of__nat__floor,axiom,
    ! [R3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R3 ) ) ) @ R3 ) ) ).

% of_nat_floor
thf(fact_8322_one__add__floor,axiom,
    ! [X2: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_8323_one__add__floor,axiom,
    ! [X2: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_8324_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_8325_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_8326_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_8327_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_8328_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_8329_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_8330_ceiling__altdef,axiom,
    ( archim7802044766580827645g_real
    = ( ^ [X: real] :
          ( if_int
          @ ( X
            = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
          @ ( archim6058952711729229775r_real @ X )
          @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_8331_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X: rat] :
          ( if_int
          @ ( X
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
          @ ( archim3151403230148437115or_rat @ X )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_8332_le__nat__floor,axiom,
    ! [X2: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A )
     => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_8333_ceiling__diff__floor__le__1,axiom,
    ! [X2: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim6058952711729229775r_real @ X2 ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_8334_ceiling__diff__floor__le__1,axiom,
    ! [X2: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim3151403230148437115or_rat @ X2 ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_8335_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_8336_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : ( ord_less_real @ R3 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_8337_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_8338_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : ( ord_less_real @ ( minus_minus_real @ R3 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_8339_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( minus_minus_real @ R3 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_8340_pochhammer__rec,axiom,
    ! [A: word_N3645301735248828278l_num1,N2: nat] :
      ( ( comm_s6431939913906641691l_num1 @ A @ ( suc @ N2 ) )
      = ( times_7065122842183080059l_num1 @ A @ ( comm_s6431939913906641691l_num1 @ ( plus_p361126936061061375l_num1 @ A @ one_on7727431528512463931l_num1 ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8341_pochhammer__rec,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8342_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8343_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8344_pochhammer__rec,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8345_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8346_pochhammer__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8347_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8348_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8349_pochhammer__Suc,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A @ N2 ) @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8350_pochhammer__rec_H,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8351_pochhammer__rec_H,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8352_pochhammer__rec_H,axiom,
    ! [Z: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8353_pochhammer__rec_H,axiom,
    ! [Z: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8354_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N2 ) ) @ ( comm_s8582702949713902594nteger @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8355_pochhammer__product_H,axiom,
    ! [Z: rat,N2: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8356_pochhammer__product_H,axiom,
    ! [Z: real,N2: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8357_pochhammer__product_H,axiom,
    ! [Z: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8358_pochhammer__product_H,axiom,
    ! [Z: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8359_pochhammer__product_H,axiom,
    ! [Z: code_integer,N2: nat,M: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ N2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8360_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8361_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8362_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8363_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8364_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N: nat] : ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8365_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8366_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8367_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N: nat] : ( minus_minus_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8368_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8369_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8370_sumr__diff__mult__const2,axiom,
    ! [F: nat > rat,N2: nat,R3: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ R3 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ I2 ) @ R3 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8371_sumr__diff__mult__const2,axiom,
    ! [F: nat > int,N2: nat,R3: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ R3 ) )
      = ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( minus_minus_int @ ( F @ I2 ) @ R3 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8372_sumr__diff__mult__const2,axiom,
    ! [F: nat > code_integer,N2: nat,R3: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( groups7501900531339628137nteger @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ R3 ) )
      = ( groups7501900531339628137nteger
        @ ^ [I2: nat] : ( minus_8373710615458151222nteger @ ( F @ I2 ) @ R3 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8373_sumr__diff__mult__const2,axiom,
    ! [F: nat > real,N2: nat,R3: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ R3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( minus_minus_real @ ( F @ I2 ) @ R3 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8374_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8375_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8376_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S3: real] :
      ( ( sums_real
        @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
        @ S3 )
      = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_8377_sums__split__initial__segment,axiom,
    ! [F: nat > real,S3: real,N2: nat] :
      ( ( sums_real @ F @ S3 )
     => ( sums_real
        @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
        @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_8378_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S3: real] :
      ( ( sums_real
        @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
        @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S3 ) ) ).

% sums_iff_shift'
thf(fact_8379_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
           => ( P @ I2 ) ) ) ) ).

% floor_split
thf(fact_8380_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
           => ( P @ I2 ) ) ) ) ).

% floor_split
thf(fact_8381_floor__eq__iff,axiom,
    ! [X2: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X2 )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X2 )
        & ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_8382_floor__eq__iff,axiom,
    ! [X2: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X2 )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X2 )
        & ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_8383_floor__unique,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X2 )
          = Z ) ) ) ).

% floor_unique
thf(fact_8384_floor__unique,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X2 )
     => ( ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X2 )
          = Z ) ) ) ).

% floor_unique
thf(fact_8385_less__floor__iff,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 ) ) ).

% less_floor_iff
thf(fact_8386_less__floor__iff,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 ) ) ).

% less_floor_iff
thf(fact_8387_floor__le__iff,axiom,
    ! [X2: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ Z )
      = ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_8388_floor__le__iff,axiom,
    ! [X2: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z )
      = ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_8389_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8390_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8391_floor__correct,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8392_floor__correct,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8393_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_8394_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_8395_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).

% floor_divide_real_eq_div
thf(fact_8396_power__diff__1__eq,axiom,
    ! [X2: uint32,N2: nat] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ X2 @ N2 ) @ one_one_uint32 )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ X2 @ one_one_uint32 ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8397_power__diff__1__eq,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: nat] :
      ( ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) @ one_on7727431528512463931l_num1 )
      = ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) @ ( groups2996710295995929986l_num1 @ ( power_2184487114949457152l_num1 @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8398_power__diff__1__eq,axiom,
    ! [X2: rat,N2: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N2 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8399_power__diff__1__eq,axiom,
    ! [X2: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8400_power__diff__1__eq,axiom,
    ! [X2: real,N2: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ N2 ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8401_one__diff__power__eq,axiom,
    ! [X2: uint32,N2: nat] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( power_power_uint32 @ X2 @ N2 ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X2 ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8402_one__diff__power__eq,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: nat] :
      ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) )
      = ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ X2 ) @ ( groups2996710295995929986l_num1 @ ( power_2184487114949457152l_num1 @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8403_one__diff__power__eq,axiom,
    ! [X2: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8404_one__diff__power__eq,axiom,
    ! [X2: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8405_one__diff__power__eq,axiom,
    ! [X2: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8406_geometric__sum,axiom,
    ! [X2: rat,N2: nat] :
      ( ( X2 != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_8407_geometric__sum,axiom,
    ! [X2: real,N2: nat] :
      ( ( X2 != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_8408_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8409_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s7457072308508201937r_real @ Z @ N2 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8410_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8411_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8412_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s8582702949713902594nteger @ Z @ N2 )
        = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8413_floor__divide__lower,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_8414_floor__divide__lower,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_8415_lemma__termdiff1,axiom,
    ! [Z: uint32,H2: uint32,M: nat] :
      ( ( groups833757482993574392uint32
        @ ^ [P5: nat] : ( minus_minus_uint32 @ ( times_times_uint32 @ ( power_power_uint32 @ ( plus_plus_uint32 @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_uint32 @ Z @ P5 ) ) @ ( power_power_uint32 @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups833757482993574392uint32
        @ ^ [P5: nat] : ( times_times_uint32 @ ( power_power_uint32 @ Z @ P5 ) @ ( minus_minus_uint32 @ ( power_power_uint32 @ ( plus_plus_uint32 @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_uint32 @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8416_lemma__termdiff1,axiom,
    ! [Z: word_N3645301735248828278l_num1,H2: word_N3645301735248828278l_num1,M: nat] :
      ( ( groups2996710295995929986l_num1
        @ ^ [P5: nat] : ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ ( plus_p361126936061061375l_num1 @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_2184487114949457152l_num1 @ Z @ P5 ) ) @ ( power_2184487114949457152l_num1 @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2996710295995929986l_num1
        @ ^ [P5: nat] : ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ Z @ P5 ) @ ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ ( plus_p361126936061061375l_num1 @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_2184487114949457152l_num1 @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8417_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8418_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8419_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8420_sum__gp__strict,axiom,
    ! [X2: rat,N2: nat] :
      ( ( ( X2 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) )
      & ( ( X2 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8421_sum__gp__strict,axiom,
    ! [X2: real,N2: nat] :
      ( ( ( X2 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ N2 ) ) )
      & ( ( X2 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8422_power__diff__sumr2,axiom,
    ! [X2: uint32,N2: nat,Y2: uint32] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ X2 @ N2 ) @ ( power_power_uint32 @ Y2 @ N2 ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ X2 @ Y2 )
        @ ( groups833757482993574392uint32
          @ ^ [I2: nat] : ( times_times_uint32 @ ( power_power_uint32 @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_uint32 @ X2 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8423_power__diff__sumr2,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: nat,Y2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) @ ( power_2184487114949457152l_num1 @ Y2 @ N2 ) )
      = ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ Y2 )
        @ ( groups2996710295995929986l_num1
          @ ^ [I2: nat] : ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_2184487114949457152l_num1 @ X2 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8424_power__diff__sumr2,axiom,
    ! [X2: rat,N2: nat,Y2: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N2 ) @ ( power_power_rat @ Y2 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y2 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( times_times_rat @ ( power_power_rat @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_rat @ X2 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8425_power__diff__sumr2,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X2 @ Y2 )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( times_times_int @ ( power_power_int @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_int @ X2 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8426_power__diff__sumr2,axiom,
    ! [X2: real,N2: nat,Y2: real] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X2 @ Y2 )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_real @ X2 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8427_diff__power__eq__sum,axiom,
    ! [X2: uint32,N2: nat,Y2: uint32] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ X2 @ ( suc @ N2 ) ) @ ( power_power_uint32 @ Y2 @ ( suc @ N2 ) ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ X2 @ Y2 )
        @ ( groups833757482993574392uint32
          @ ^ [P5: nat] : ( times_times_uint32 @ ( power_power_uint32 @ X2 @ P5 ) @ ( power_power_uint32 @ Y2 @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8428_diff__power__eq__sum,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: nat,Y2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ ( suc @ N2 ) ) @ ( power_2184487114949457152l_num1 @ Y2 @ ( suc @ N2 ) ) )
      = ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ Y2 )
        @ ( groups2996710295995929986l_num1
          @ ^ [P5: nat] : ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ P5 ) @ ( power_2184487114949457152l_num1 @ Y2 @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8429_diff__power__eq__sum,axiom,
    ! [X2: rat,N2: nat,Y2: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) @ ( power_power_rat @ Y2 @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y2 )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X2 @ P5 ) @ ( power_power_rat @ Y2 @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8430_diff__power__eq__sum,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ ( suc @ N2 ) ) @ ( power_power_int @ Y2 @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X2 @ Y2 )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X2 @ P5 ) @ ( power_power_int @ Y2 @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8431_diff__power__eq__sum,axiom,
    ! [X2: real,N2: nat,Y2: real] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ ( suc @ N2 ) ) @ ( power_power_real @ Y2 @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X2 @ Y2 )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X2 @ P5 ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8432_floor__divide__upper,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ P4 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).

% floor_divide_upper
thf(fact_8433_floor__divide__upper,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).

% floor_divide_upper
thf(fact_8434_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8435_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8436_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > code_integer,K6: code_integer,K: nat] :
      ( ! [P8: nat] :
          ( ( ord_less_nat @ P8 @ N2 )
         => ( ord_le3102999989581377725nteger @ ( F @ P8 ) @ K6 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ K6 )
       => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8437_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > rat,K6: rat,K: nat] :
      ( ! [P8: nat] :
          ( ( ord_less_nat @ P8 @ N2 )
         => ( ord_less_eq_rat @ ( F @ P8 ) @ K6 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K6 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8438_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > int,K6: int,K: nat] :
      ( ! [P8: nat] :
          ( ( ord_less_nat @ P8 @ N2 )
         => ( ord_less_eq_int @ ( F @ P8 ) @ K6 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K6 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8439_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > nat,K6: nat,K: nat] :
      ( ! [P8: nat] :
          ( ( ord_less_nat @ P8 @ N2 )
         => ( ord_less_eq_nat @ ( F @ P8 ) @ K6 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K6 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8440_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > real,K6: real,K: nat] :
      ( ! [P8: nat] :
          ( ( ord_less_nat @ P8 @ N2 )
         => ( ord_less_eq_real @ ( F @ P8 ) @ K6 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K6 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8441_one__diff__power__eq_H,axiom,
    ! [X2: uint32,N2: nat] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( power_power_uint32 @ X2 @ N2 ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X2 )
        @ ( groups833757482993574392uint32
          @ ^ [I2: nat] : ( power_power_uint32 @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8442_one__diff__power__eq_H,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: nat] :
      ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) )
      = ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ X2 )
        @ ( groups2996710295995929986l_num1
          @ ^ [I2: nat] : ( power_2184487114949457152l_num1 @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8443_one__diff__power__eq_H,axiom,
    ! [X2: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( power_power_rat @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8444_one__diff__power__eq_H,axiom,
    ! [X2: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( power_power_int @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8445_one__diff__power__eq_H,axiom,
    ! [X2: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8446_TBOUND__adm,axiom,
    ! [T: produc3881548065746020326Ti_nat > nat] :
      ( comple6931689918642796574on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: produc3881548065746020326Ti_nat] : ( time_T8353473612707095248on_nat @ ( F4 @ X ) @ ( T @ X ) ) ) ).

% TBOUND_adm
thf(fact_8447_TBOUND__adm,axiom,
    ! [T: produc3881548065746020326Ti_nat > nat] :
      ( comple6491863954676465222Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
        ! [X: produc3881548065746020326Ti_nat] : ( time_TBOUND_o @ ( F4 @ X ) @ ( T @ X ) ) ) ).

% TBOUND_adm
thf(fact_8448_TBOUND__adm,axiom,
    ! [T: produc3960855945107176009Ti_nat > nat] :
      ( comple1745167176254620304_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
        ! [X: produc3960855945107176009Ti_nat] : ( time_T5737551269749752165_VEBTi @ ( F4 @ X ) @ ( T @ X ) ) ) ).

% TBOUND_adm
thf(fact_8449_TBOUND__adm,axiom,
    ! [T: produc3960855945107176009Ti_nat > nat] :
      ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: produc3960855945107176009Ti_nat] : ( time_T8353473612707095248on_nat @ ( F4 @ X ) @ ( T @ X ) ) ) ).

% TBOUND_adm
thf(fact_8450_TBOUND__adm,axiom,
    ! [T: produc3960855945107176009Ti_nat > nat] :
      ( comple2654586775044187945Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
        ! [X: produc3960855945107176009Ti_nat] : ( time_TBOUND_o @ ( F4 @ X ) @ ( T @ X ) ) ) ).

% TBOUND_adm
thf(fact_8451_refines__adm,axiom,
    ! [T: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
      ( comple6931689918642796574on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: produc3881548065746020326Ti_nat] : ( refine7594492741263601813on_nat @ ( T @ X ) @ ( F4 @ X ) ) ) ).

% refines_adm
thf(fact_8452_refines__adm,axiom,
    ! [T: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
      ( comple6491863954676465222Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
        ! [X: produc3881548065746020326Ti_nat] : ( refine_Imp_refines_o @ ( T @ X ) @ ( F4 @ X ) ) ) ).

% refines_adm
thf(fact_8453_refines__adm,axiom,
    ! [T: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
      ( comple1745167176254620304_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
        ! [X: produc3960855945107176009Ti_nat] : ( refine5565527176597971370_VEBTi @ ( T @ X ) @ ( F4 @ X ) ) ) ).

% refines_adm
thf(fact_8454_refines__adm,axiom,
    ! [T: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
      ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: produc3960855945107176009Ti_nat] : ( refine7594492741263601813on_nat @ ( T @ X ) @ ( F4 @ X ) ) ) ).

% refines_adm
thf(fact_8455_refines__adm,axiom,
    ! [T: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
      ( comple2654586775044187945Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
        ! [X: produc3960855945107176009Ti_nat] : ( refine_Imp_refines_o @ ( T @ X ) @ ( F4 @ X ) ) ) ).

% refines_adm
thf(fact_8456_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_8457_floor__log__eq__powr__iff,axiom,
    ! [X2: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X2 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X2 )
            & ( ord_less_real @ X2 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_8458_pochhammer__code,axiom,
    ( comm_s6431939913906641691l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,N: nat] :
          ( if_wor5778924947035936048l_num1 @ ( N = zero_zero_nat ) @ one_on7727431528512463931l_num1
          @ ( set_fo4709898541180519304l_num1
            @ ^ [O: nat] : ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ A4 @ ( semiri8819519690708144855l_num1 @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_on7727431528512463931l_num1 ) ) ) ) ).

% pochhammer_code
thf(fact_8459_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_8460_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_8461_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_8462_pochhammer__code,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A4: code_integer,N: nat] :
          ( if_Code_integer @ ( N = zero_zero_nat ) @ one_one_Code_integer
          @ ( set_fo1084959871951514735nteger
            @ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A4 @ ( semiri4939895301339042750nteger @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_Code_integer ) ) ) ) ).

% pochhammer_code
thf(fact_8463_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_8464_TBOUND__fi__adm,axiom,
    ! [Foo: vEBT_VEBTi > nat > nat] :
      ( comple6931689918642796574on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [Fi: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: vEBT_VEBTi,Xa4: nat] : ( time_T8353473612707095248on_nat @ ( produc1489253303066280154on_nat @ Fi @ X @ Xa4 ) @ ( Foo @ X @ Xa4 ) ) ) ).

% TBOUND_fi_adm
thf(fact_8465_TBOUND__fi__adm,axiom,
    ! [Foo: vEBT_VEBTi > nat > nat] :
      ( comple6491863954676465222Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [Fi: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
        ! [X: vEBT_VEBTi,Xa4: nat] : ( time_TBOUND_o @ ( produc5685940877448195828Heap_o @ Fi @ X @ Xa4 ) @ ( Foo @ X @ Xa4 ) ) ) ).

% TBOUND_fi_adm
thf(fact_8466_TBOUND__fi__adm,axiom,
    ! [Foo: produc3625547720036274456_VEBTi > nat > nat] :
      ( comple1745167176254620304_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi )
      @ ^ [Fi: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
        ! [X: produc3625547720036274456_VEBTi,Xa4: nat] : ( time_T5737551269749752165_VEBTi @ ( produc2164094337957399884_VEBTi @ Fi @ X @ Xa4 ) @ ( Foo @ X @ Xa4 ) ) ) ).

% TBOUND_fi_adm
thf(fact_8467_TBOUND__fi__adm,axiom,
    ! [Foo: produc3625547720036274456_VEBTi > nat > nat] :
      ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
      @ ^ [Fi: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
        ! [X: produc3625547720036274456_VEBTi,Xa4: nat] : ( time_T8353473612707095248on_nat @ ( produc1757988346207259447on_nat @ Fi @ X @ Xa4 ) @ ( Foo @ X @ Xa4 ) ) ) ).

% TBOUND_fi_adm
thf(fact_8468_TBOUND__fi__adm,axiom,
    ! [Foo: produc3625547720036274456_VEBTi > nat > nat] :
      ( comple2654586775044187945Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o )
      @ ^ [Fi: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
        ! [X: produc3625547720036274456_VEBTi,Xa4: nat] : ( time_TBOUND_o @ ( produc8381543706267210711Heap_o @ Fi @ X @ Xa4 ) @ ( Foo @ X @ Xa4 ) ) ) ).

% TBOUND_fi_adm
thf(fact_8469_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_8470_floor__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_8471_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8472_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8473_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K4: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8474_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N2: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K4: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8475_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M7: nat] : ( times_times_real @ ( cos_coeff @ M7 ) @ ( power_power_real @ zero_zero_real @ M7 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8476_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).

% finite_nat_bounded
thf(fact_8477_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S8: set_nat] :
        ? [K4: nat] : ( ord_less_eq_set_nat @ S8 @ ( set_ord_lessThan_nat @ K4 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_8478_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_8479_prod_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_8480_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A2 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_8481_of__nat__prod,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8482_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8483_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [X: nat] : ( semiri4939895301339042750nteger @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8484_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8485_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8486_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8487_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [X: nat] : ( ring_1_of_int_rat @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8488_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( ring_1_of_int_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8489_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups2316167850115554303t_real
        @ ^ [X: int] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8490_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1072433553688619179nt_rat
        @ ^ [X: int] : ( ring_1_of_int_rat @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8491_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( ring_1_of_int_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8492_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_8493_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_8494_prod_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
      = one_one_real ) ).

% prod.empty
thf(fact_8495_prod_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
      = one_one_rat ) ).

% prod.empty
thf(fact_8496_prod_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
      = one_one_nat ) ).

% prod.empty
thf(fact_8497_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_8498_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_8499_prod_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty
thf(fact_8500_prod_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty
thf(fact_8501_prod_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups705719431365010083at_int @ G @ bot_bot_set_nat )
      = one_one_int ) ).

% prod.empty
thf(fact_8502_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8503_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8504_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8505_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: code_integer > real] :
      ( ~ ( finite6017078050557962740nteger @ A2 )
     => ( ( groups9004974159866482096r_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8506_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8507_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8508_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8509_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat] :
      ( ~ ( finite6017078050557962740nteger @ A2 )
     => ( ( groups2555765274223993564er_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8510_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_8511_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_8512_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8513_prod_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8514_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K4: real] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K4: real] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8515_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K4: nat] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K4: nat] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8516_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K4: int] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K4: int] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8517_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K4: complex] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K4: complex] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8518_prod_Odelta_H,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups9004974159866482096r_real
              @ ^ [K4: code_integer] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups9004974159866482096r_real
              @ ^ [K4: code_integer] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8519_prod_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups5726676334696518183BT_rat
              @ ^ [K4: vEBT_VEBT] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups5726676334696518183BT_rat
              @ ^ [K4: vEBT_VEBT] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_8520_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K4: real] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K4: real] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_8521_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_8522_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K4: int] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K4: int] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_8523_prod_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K4: vEBT_VEBT] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8524_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K4: real] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K4: real] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8525_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K4: nat] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K4: nat] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8526_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K4: int] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K4: int] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8527_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K4: complex] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K4: complex] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8528_prod_Odelta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups9004974159866482096r_real
              @ ^ [K4: code_integer] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups9004974159866482096r_real
              @ ^ [K4: code_integer] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8529_prod_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups5726676334696518183BT_rat
              @ ^ [K4: vEBT_VEBT] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups5726676334696518183BT_rat
              @ ^ [K4: vEBT_VEBT] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_8530_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K4: real] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K4: real] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_8531_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_8532_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_8533_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
          = ( times_times_real @ ( G @ X2 ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8534_prod_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X2 @ A2 )
       => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X2 @ A2 ) )
          = ( times_times_real @ ( G @ X2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8535_prod_Oinsert,axiom,
    ! [A2: set_nat,X2: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X2 @ A2 )
       => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X2 @ A2 ) )
          = ( times_times_real @ ( G @ X2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8536_prod_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X2 @ A2 )
       => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X2 @ A2 ) )
          = ( times_times_real @ ( G @ X2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8537_prod_Oinsert,axiom,
    ! [A2: set_complex,X2: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X2 @ A2 )
       => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X2 @ A2 ) )
          = ( times_times_real @ ( G @ X2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8538_prod_Oinsert,axiom,
    ! [A2: set_Code_integer,X2: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X2 @ A2 )
       => ( ( groups9004974159866482096r_real @ G @ ( insert_Code_integer @ X2 @ A2 ) )
          = ( times_times_real @ ( G @ X2 ) @ ( groups9004974159866482096r_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8539_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
          = ( times_times_rat @ ( G @ X2 ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8540_prod_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X2 @ A2 )
       => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X2 @ A2 ) )
          = ( times_times_rat @ ( G @ X2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8541_prod_Oinsert,axiom,
    ! [A2: set_nat,X2: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X2 @ A2 )
       => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X2 @ A2 ) )
          = ( times_times_rat @ ( G @ X2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8542_prod_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X2 @ A2 )
       => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X2 @ A2 ) )
          = ( times_times_rat @ ( G @ X2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8543_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8544_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8545_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8546_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8547_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > word_N3645301735248828278l_num1] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups5040993209112964077l_num1 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_on7727431528512463931l_num1 ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups5040993209112964077l_num1 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_7065122842183080059l_num1 @ ( groups5040993209112964077l_num1 @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8548_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8549_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8550_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8551_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8552_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A2: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8553_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > real,A2: set_VEBT_VEBT] :
      ( ( ( groups2703838992350267259T_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8554_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8555_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8556_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups73079841787564623at_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8557_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > rat,A2: set_VEBT_VEBT] :
      ( ( ( groups5726676334696518183BT_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8558_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups4061424788464935467al_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8559_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A2: set_int] :
      ( ( ( groups1072433553688619179nt_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8560_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > nat,A2: set_VEBT_VEBT] :
      ( ( ( groups6361806394783013919BT_nat @ G @ A2 )
       != one_one_nat )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8561_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > nat,A2: set_real] :
      ( ( ( groups4696554848551431203al_nat @ G @ A2 )
       != one_one_nat )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_8562_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_8563_prod_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_8564_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_8565_prod_Oswap,axiom,
    ! [G: nat > nat > int,B5: set_nat,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( groups705719431365010083at_int @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I2: nat] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% prod.swap
thf(fact_8566_prod_Oswap,axiom,
    ! [G: nat > int > int,B5: set_int,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( groups1705073143266064639nt_int @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups1705073143266064639nt_int
        @ ^ [J3: int] :
            ( groups705719431365010083at_int
            @ ^ [I2: nat] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% prod.swap
thf(fact_8567_prod_Oswap,axiom,
    ! [G: int > nat > int,B5: set_nat,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I2: int] : ( groups705719431365010083at_int @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups1705073143266064639nt_int
            @ ^ [I2: int] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% prod.swap
thf(fact_8568_prod_Oswap,axiom,
    ! [G: int > int > int,B5: set_int,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I2: int] : ( groups1705073143266064639nt_int @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups1705073143266064639nt_int
        @ ^ [J3: int] :
            ( groups1705073143266064639nt_int
            @ ^ [I2: int] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% prod.swap
thf(fact_8569_prod_Oswap,axiom,
    ! [G: nat > nat > nat,B5: set_nat,A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( groups708209901874060359at_nat @ ( G @ I2 ) @ B5 )
        @ A2 )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I2: nat] : ( G @ I2 @ J3 )
            @ A2 )
        @ B5 ) ) ).

% prod.swap
thf(fact_8570_prod_Odistrib,axiom,
    ! [G: nat > int,H2: nat > int,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_8571_prod_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_8572_prod_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_8573_prod__power__distrib,axiom,
    ! [F: nat > int,A2: set_nat,N2: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N2 )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_8574_prod__power__distrib,axiom,
    ! [F: int > int,A2: set_int,N2: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N2 )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_8575_prod__power__distrib,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N2 )
      = ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_8576_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_nat,G: vEBT_VEBT > nat > int,R2: vEBT_VEBT > nat > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups6359315924273963643BT_int
            @ ^ [X: vEBT_VEBT] :
                ( groups705719431365010083at_int @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y: nat] :
                ( groups6359315924273963643BT_int
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8577_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B5: set_nat,G: real > nat > int,R2: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups4694064378042380927al_int
            @ ^ [X: real] :
                ( groups705719431365010083at_int @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y: nat] :
                ( groups4694064378042380927al_int
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8578_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B5: set_nat,G: complex > nat > int,R2: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups858564598930262913ex_int
            @ ^ [X: complex] :
                ( groups705719431365010083at_int @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y: nat] :
                ( groups858564598930262913ex_int
                @ ^ [X: complex] : ( G @ X @ Y )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8579_prod_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B5: set_nat,G: code_integer > nat > int,R2: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3188404863801439024er_int
            @ ^ [X: code_integer] :
                ( groups705719431365010083at_int @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y: nat] :
                ( groups3188404863801439024er_int
                @ ^ [X: code_integer] : ( G @ X @ Y )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8580_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_int,G: vEBT_VEBT > int > int,R2: vEBT_VEBT > int > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups6359315924273963643BT_int
            @ ^ [X: vEBT_VEBT] :
                ( groups1705073143266064639nt_int @ ( G @ X )
                @ ( collect_int
                  @ ^ [Y: int] :
                      ( ( member_int @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y: int] :
                ( groups6359315924273963643BT_int
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8581_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B5: set_int,G: real > int > int,R2: real > int > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups4694064378042380927al_int
            @ ^ [X: real] :
                ( groups1705073143266064639nt_int @ ( G @ X )
                @ ( collect_int
                  @ ^ [Y: int] :
                      ( ( member_int @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y: int] :
                ( groups4694064378042380927al_int
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8582_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B5: set_int,G: complex > int > int,R2: complex > int > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups858564598930262913ex_int
            @ ^ [X: complex] :
                ( groups1705073143266064639nt_int @ ( G @ X )
                @ ( collect_int
                  @ ^ [Y: int] :
                      ( ( member_int @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y: int] :
                ( groups858564598930262913ex_int
                @ ^ [X: complex] : ( G @ X @ Y )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8583_prod_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B5: set_int,G: code_integer > int > int,R2: code_integer > int > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups3188404863801439024er_int
            @ ^ [X: code_integer] :
                ( groups1705073143266064639nt_int @ ( G @ X )
                @ ( collect_int
                  @ ^ [Y: int] :
                      ( ( member_int @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y: int] :
                ( groups3188404863801439024er_int
                @ ^ [X: code_integer] : ( G @ X @ Y )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8584_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_nat,G: vEBT_VEBT > nat > nat,R2: vEBT_VEBT > nat > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups6361806394783013919BT_nat
            @ ^ [X: vEBT_VEBT] :
                ( groups708209901874060359at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y: nat] :
                ( groups6361806394783013919BT_nat
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8585_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B5: set_nat,G: real > nat > nat,R2: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups4696554848551431203al_nat
            @ ^ [X: real] :
                ( groups708209901874060359at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( R2 @ X @ Y ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y: nat] :
                ( groups4696554848551431203al_nat
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R2 @ X @ Y ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8586_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A2: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I2: nat] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8587_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I2: int] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8588_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I2: nat] : ( modulo_modulo_nat @ ( F @ I2 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8589_prod__mono,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8590_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups2703838992350267259T_real @ F @ A2 ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8591_prod__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8592_prod__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8593_prod__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8594_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8595_prod__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8596_prod__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8597_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) @ ( groups6361806394783013919BT_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8598_prod__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_8599_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_8600_prod__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_8601_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_8602_prod__pos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_8603_prod__pos,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_8604_prod__pos,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_8605_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8606_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2703838992350267259T_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8607_prod__ge__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8608_prod__ge__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8609_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8610_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8611_prod__ge__1,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8612_prod__ge__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8613_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8614_prod__ge__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_8615_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2703838992350267259T_real @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups2703838992350267259T_real
          @ ^ [X: vEBT_VEBT] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8616_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups1681761925125756287l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8617_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > real,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups129246275422532515t_real
          @ ^ [X: nat] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8618_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups2316167850115554303t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8619_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups766887009212190081x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8620_prod_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: code_integer > real,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups9004974159866482096r_real @ G
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ( ( member_Code_integer @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups9004974159866482096r_real
          @ ^ [X: code_integer] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8621_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > rat,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups5726676334696518183BT_rat @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups5726676334696518183BT_rat
          @ ^ [X: vEBT_VEBT] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8622_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4061424788464935467al_rat @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups4061424788464935467al_rat
          @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8623_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups73079841787564623at_rat
          @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8624_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups1072433553688619179nt_rat
          @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_8625_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_8626_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_8627_power__sum,axiom,
    ! [C: uint32,F: nat > nat,A2: set_nat] :
      ( ( power_power_uint32 @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups2278496514549435363uint32
        @ ^ [A4: nat] : ( power_power_uint32 @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_8628_power__sum,axiom,
    ! [C: word_N3645301735248828278l_num1,F: nat > nat,A2: set_nat] :
      ( ( power_2184487114949457152l_num1 @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups5040993209112964077l_num1
        @ ^ [A4: nat] : ( power_2184487114949457152l_num1 @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_8629_power__sum,axiom,
    ! [C: real,F: nat > nat,A2: set_nat] :
      ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [A4: nat] : ( power_power_real @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_8630_power__sum,axiom,
    ! [C: int,F: nat > nat,A2: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [A4: nat] : ( power_power_int @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_8631_power__sum,axiom,
    ! [C: int,F: int > nat,A2: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A4: int] : ( power_power_int @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_8632_power__sum,axiom,
    ! [C: nat,F: nat > nat,A2: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A4: nat] : ( power_power_nat @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_8633_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_8634_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_8635_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8636_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2703838992350267259T_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8637_prod__le__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8638_prod__le__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8639_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8640_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8641_prod__le__1,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8642_prod__le__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8643_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8644_prod__le__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8645_prod_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X16: real,Y15: real,X24: real,Y24: real] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8646_prod_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X16: real,Y15: real,X24: real,Y24: real] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8647_prod_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X16: real,Y15: real,X24: real,Y24: real] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8648_prod_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X16: real,Y15: real,X24: real,Y24: real] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups9004974159866482096r_real @ H2 @ S2 ) @ ( groups9004974159866482096r_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8649_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X16: rat,Y15: rat,X24: rat,Y24: rat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8650_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X16: rat,Y15: rat,X24: rat,Y24: rat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8651_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X16: rat,Y15: rat,X24: rat,Y24: rat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8652_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_Code_integer,H2: code_integer > rat,G: code_integer > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X16: rat,Y15: rat,X24: rat,Y24: rat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X24 @ Y24 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups2555765274223993564er_rat @ H2 @ S2 ) @ ( groups2555765274223993564er_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8653_prod_Orelated,axiom,
    ! [R2: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
      ( ( R2 @ one_one_nat @ one_one_nat )
     => ( ! [X16: nat,Y15: nat,X24: nat,Y24: nat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8654_prod_Orelated,axiom,
    ! [R2: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R2 @ one_one_nat @ one_one_nat )
     => ( ! [X16: nat,Y15: nat,X24: nat,Y24: nat] :
            ( ( ( R2 @ X16 @ X24 )
              & ( R2 @ Y15 @ Y24 ) )
           => ( R2 @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups861055069439313189ex_nat @ H2 @ S2 ) @ ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8655_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( groups2703838992350267259T_real @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( times_times_real @ ( G @ X2 ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8656_prod_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X2 @ A2 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X2 @ A2 ) )
            = ( groups1681761925125756287l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X2 @ A2 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X2 @ A2 ) )
            = ( times_times_real @ ( G @ X2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8657_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X2: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X2 @ A2 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X2 @ A2 ) )
            = ( groups129246275422532515t_real @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X2 @ A2 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X2 @ A2 ) )
            = ( times_times_real @ ( G @ X2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8658_prod_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X2 @ A2 )
         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X2 @ A2 ) )
            = ( groups2316167850115554303t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X2 @ A2 )
         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X2 @ A2 ) )
            = ( times_times_real @ ( G @ X2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8659_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X2: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X2 @ A2 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X2 @ A2 ) )
            = ( groups766887009212190081x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X2 @ A2 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X2 @ A2 ) )
            = ( times_times_real @ ( G @ X2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8660_prod_Oinsert__if,axiom,
    ! [A2: set_Code_integer,X2: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ X2 @ A2 )
         => ( ( groups9004974159866482096r_real @ G @ ( insert_Code_integer @ X2 @ A2 ) )
            = ( groups9004974159866482096r_real @ G @ A2 ) ) )
        & ( ~ ( member_Code_integer @ X2 @ A2 )
         => ( ( groups9004974159866482096r_real @ G @ ( insert_Code_integer @ X2 @ A2 ) )
            = ( times_times_real @ ( G @ X2 ) @ ( groups9004974159866482096r_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8661_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( groups5726676334696518183BT_rat @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X2 @ A2 )
         => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X2 @ A2 ) )
            = ( times_times_rat @ ( G @ X2 ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8662_prod_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X2 @ A2 )
         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X2 @ A2 ) )
            = ( groups4061424788464935467al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X2 @ A2 )
         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X2 @ A2 ) )
            = ( times_times_rat @ ( G @ X2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8663_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X2: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X2 @ A2 )
         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X2 @ A2 ) )
            = ( groups73079841787564623at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X2 @ A2 )
         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X2 @ A2 ) )
            = ( times_times_rat @ ( G @ X2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8664_prod_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X2 @ A2 )
         => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X2 @ A2 ) )
            = ( groups1072433553688619179nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X2 @ A2 )
         => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X2 @ A2 ) )
            = ( times_times_rat @ ( G @ X2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_8665_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: vEBT_VEBT > vEBT_VEBT,J: vEBT_VEBT > vEBT_VEBT,T5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups2703838992350267259T_real @ G @ S2 )
                        = ( groups2703838992350267259T_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8666_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_real,S2: set_VEBT_VEBT,I: real > vEBT_VEBT,J: vEBT_VEBT > real,T5: set_real,G: vEBT_VEBT > real,H2: real > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_real @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T5 @ T7 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups2703838992350267259T_real @ G @ S2 )
                        = ( groups1681761925125756287l_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8667_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_VEBT_VEBT,S2: set_real,I: vEBT_VEBT > real,J: real > vEBT_VEBT,T5: set_VEBT_VEBT,G: real > real,H2: vEBT_VEBT > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1681761925125756287l_real @ G @ S2 )
                        = ( groups2703838992350267259T_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8668_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_real,S2: set_real,I: real > real,J: real > real,T5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_real @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T5 @ T7 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1681761925125756287l_real @ G @ S2 )
                        = ( groups1681761925125756287l_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8669_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_int,S2: set_VEBT_VEBT,I: int > vEBT_VEBT,J: vEBT_VEBT > int,T5: set_int,G: vEBT_VEBT > real,H2: int > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_int @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T7 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups2703838992350267259T_real @ G @ S2 )
                        = ( groups2316167850115554303t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8670_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_int,S2: set_real,I: int > real,J: real > int,T5: set_int,G: real > real,H2: int > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_int @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T7 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1681761925125756287l_real @ G @ S2 )
                        = ( groups2316167850115554303t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8671_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_complex,S2: set_VEBT_VEBT,I: complex > vEBT_VEBT,J: vEBT_VEBT > complex,T5: set_complex,G: vEBT_VEBT > real,H2: complex > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite3207457112153483333omplex @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T7 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups2703838992350267259T_real @ G @ S2 )
                        = ( groups766887009212190081x_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8672_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_complex,S2: set_real,I: complex > real,J: real > complex,T5: set_complex,G: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite3207457112153483333omplex @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T7 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1681761925125756287l_real @ G @ S2 )
                        = ( groups766887009212190081x_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8673_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T7: set_Code_integer,S2: set_VEBT_VEBT,I: code_integer > vEBT_VEBT,J: vEBT_VEBT > code_integer,T5: set_Code_integer,G: vEBT_VEBT > real,H2: code_integer > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite6017078050557962740nteger @ T7 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T5 @ T7 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups2703838992350267259T_real @ G @ S2 )
                        = ( groups9004974159866482096r_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8674_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T7: set_Code_integer,S2: set_real,I: code_integer > real,J: real > code_integer,T5: set_Code_integer,G: real > real,H2: code_integer > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite6017078050557962740nteger @ T7 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T5 @ T7 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T7 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S2 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T7 )
                       => ( ( H2 @ B2 )
                          = one_one_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S2 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1681761925125756287l_real @ G @ S2 )
                        = ( groups9004974159866482096r_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8675_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8676_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A2 ) @ ( groups3190895334310489300er_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8677_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8678_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A2 ) @ ( groups3188404863801439024er_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8679_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > uint32] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( dvd_dvd_uint32 @ ( groups2278496514549435363uint32 @ F @ A2 ) @ ( groups2278496514549435363uint32 @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8680_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > uint32] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( dvd_dvd_uint32 @ ( groups6230475983024736193uint32 @ F @ A2 ) @ ( groups6230475983024736193uint32 @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8681_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( dvd_dvd_uint32 @ ( groups5586078468126652656uint32 @ F @ A2 ) @ ( groups5586078468126652656uint32 @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8682_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8683_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8684_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( dvd_dvd_Code_integer @ ( groups3674199335183972705nteger @ F @ A2 ) @ ( groups3674199335183972705nteger @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8685_prod__dvd__prod__subset2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) @ ( groups6361806394783013919BT_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8686_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8687_prod__dvd__prod__subset2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8688_prod__dvd__prod__subset2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A2 ) @ ( groups3190895334310489300er_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8689_prod__dvd__prod__subset2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups6359315924273963643BT_int @ F @ A2 ) @ ( groups6359315924273963643BT_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8690_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A2: set_real,F: real > int,G: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8691_prod__dvd__prod__subset2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8692_prod__dvd__prod__subset2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A2 ) @ ( groups3188404863801439024er_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8693_prod__dvd__prod__subset2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > uint32,G: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ A2 )
             => ( dvd_dvd_uint32 @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_uint32 @ ( groups8305177534072719291uint32 @ F @ A2 ) @ ( groups8305177534072719291uint32 @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8694_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A2: set_real,F: real > uint32,G: real > uint32] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_uint32 @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_uint32 @ ( groups1111744456595050943uint32 @ F @ A2 ) @ ( groups1111744456595050943uint32 @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8695_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups705719431365010083at_int @ G
          @ ( minus_minus_set_nat @ A2
            @ ( collect_nat
              @ ^ [X: nat] :
                  ( ( G @ X )
                  = one_one_int ) ) ) )
        = ( groups705719431365010083at_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8696_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1705073143266064639nt_int @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = one_one_int ) ) ) )
        = ( groups1705073143266064639nt_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8697_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups708209901874060359at_nat @ G
          @ ( minus_minus_set_nat @ A2
            @ ( collect_nat
              @ ^ [X: nat] :
                  ( ( G @ X )
                  = one_one_nat ) ) ) )
        = ( groups708209901874060359at_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8698_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M2: nat] :
          ( ( ord_less_nat @ K @ M2 )
         => ? [N8: nat] :
              ( ( ord_less_nat @ M2 @ N8 )
              & ( member_nat @ N8 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_8699_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M7: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M7 @ N )
            & ( member_nat @ N @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_8700_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M7: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M7 @ N )
            & ( member_nat @ N @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_8701_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_8702_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_8703_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_8704_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_8705_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_8706_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_8707_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_8708_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_8709_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_8710_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ one_one_nat )
      = N2 ) ).

% choose_one
thf(fact_8711_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8712_Suc__times__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_8713_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_8714_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_8715_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R3 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_8716_binomial__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R3 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R3 ) @ ( power_power_nat @ N2 @ R3 ) ) ) ).

% binomial_le_pow
thf(fact_8717_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_8718_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_8719_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_8720_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_8721_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_8722_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_8723_small__lazy_H_Ocases,axiom,
    ! [X2: product_prod_int_int] :
      ~ ! [D3: int,I3: int] :
          ( X2
         != ( product_Pair_int_int @ D3 @ I3 ) ) ).

% small_lazy'.cases
thf(fact_8724_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_8725_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_8726_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_8727_binomial__antimono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K2 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K2 @ N2 )
         => ( ord_less_eq_nat @ ( binomial @ N2 @ K2 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_8728_binomial__mono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ N2 )
       => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K2 ) ) ) ) ).

% binomial_mono
thf(fact_8729_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_8730_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_8731_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_8732_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_8733_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_8734_binomial__strict__mono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ N2 )
       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K2 ) ) ) ) ).

% binomial_strict_mono
thf(fact_8735_binomial__strict__antimono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K2 )
     => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K2 @ N2 )
         => ( ord_less_nat @ ( binomial @ N2 @ K2 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_8736_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_8737_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_8738_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_8739_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_8740_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_8741_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S8: set_nat] :
        ? [K4: nat] : ( ord_less_eq_set_nat @ S8 @ ( set_ord_atMost_nat @ K4 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_8742_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( binomial @ K4 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_8743_sum__choose__lower,axiom,
    ! [R3: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K4 ) @ K4 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_8744_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_8745_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_8746_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K4: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K4 ) @ ( minus_minus_nat @ M @ K4 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_8747_vandermonde,axiom,
    ! [M: nat,N2: nat,R3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( times_times_nat @ ( binomial @ M @ K4 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R3 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ R3 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R3 ) ) ).

% vandermonde
thf(fact_8748_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_8749_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_8750_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K4 ) ) @ ( power_power_nat @ A @ K4 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_8751_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X2: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X2 @ I2 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K4: nat] : ( times_times_nat @ ( A @ K4 ) @ ( B @ ( minus_minus_nat @ R @ K4 ) ) )
                  @ ( set_ord_atMost_nat @ R ) )
                @ ( power_power_nat @ X2 @ R ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8752_choose__square__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( power_power_nat @ ( binomial @ N2 @ K4 ) @ ( 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_8753_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_8754_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N2 @ I2 ) )
        @ ( 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_8755_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_8756_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y2
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y2
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y2
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y2
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                     => ( ~ Y2
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_8757_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_8758_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_8759_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_8760_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_8761_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ( divide_divide_int @ A @ B )
          = ( uminus_uminus_int @ A ) )
        = ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_div_minus_is_minus1
thf(fact_8762_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_8763_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_8764_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_8765_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_8766_int__cases,axiom,
    ! [Z: int] :
      ( ! [N4: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% int_cases
thf(fact_8767_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
     => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_8768_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_8769_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_8770_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_8771_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_8772_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_8773_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_8774_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_8775_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_8776_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_8777_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_8778_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_8779_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_8780_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R3 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R3 ) ) ) @ ( power_power_nat @ N2 @ R3 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8781_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_8782_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_8783_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_8784_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_8785_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_8786_negD,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ zero_zero_int )
     => ? [N4: nat] :
          ( X2
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% negD
thf(fact_8787_verit__less__mono__div__int2,axiom,
    ! [A2: int,B5: int,N2: int] :
      ( ( ord_less_eq_int @ A2 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_8788_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_8789_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_8790_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_8791_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_8792_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_8793_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_8794_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_8795_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_8796_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_8797_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_8798_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R3 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R3 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R3 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_8799_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_8800_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_8801_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K3: int] :
              ( ( P @ K3 )
             => ( ( K3 != zero_zero_int )
               => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K3: int] :
                ( ( P @ K3 )
               => ( ( K3
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_8802_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_8803_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
              @ ^ [M7: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M7 ) @ ( semiri2265585572941072030t_real @ M7 ) ) @ ( power_power_real @ H2 @ M7 ) )
              @ ( set_ord_lessThan_nat @ N2 ) )
            @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8804_sb__dec__lem_H,axiom,
    ! [K: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( 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 ) ) @ A ) ) ) ).

% sb_dec_lem'
thf(fact_8805_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_8806_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ! [Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_8807_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_8808_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A @ ( 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 @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem'
thf(fact_8809_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( 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 ) ) @ A ) ) ) ).

% sb_dec_lem
thf(fact_8810_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K4: nat] : ( if_nat @ ( ord_less_nat @ N @ K4 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K4 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K4 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K4 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8811_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_8812_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_8813_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K3: int,L3: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L3 ) )
           => ( ( ~ ( ( member_int @ K3 @ ( 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 @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K3 @ L3 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_8814_real__add__minus__iff,axiom,
    ! [X2: real,A: real] :
      ( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X2 = A ) ) ).

% real_add_minus_iff
thf(fact_8815_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_8816_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_8817_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_8818_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_8819_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_8820_real__minus__mult__self__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).

% real_minus_mult_self_le
thf(fact_8821_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).

% minus_real_def
thf(fact_8822_complex__mod__minus__le__complex__mod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_8823_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_8824_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_8825_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_8826_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_8827_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_8828_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
      = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).

% divide_powr_uminus
thf(fact_8829_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_8830_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_8831_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L2: int,R: int] : ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_8832_Bernoulli__inequality,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N2 ) ) ) ).

% Bernoulli_inequality
thf(fact_8833_log__minus__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ ( log @ B @ X2 ) @ Y2 )
            = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ ( uminus_uminus_real @ Y2 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8834_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_8835_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_8836_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_8837_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_8838_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_8839_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_8840_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_8841_sin__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( sin_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_8842_cos__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( cos_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_8843_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K4: int] :
          ( if_Code_integer @ ( ord_less_int @ K4 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K4 ) ) )
          @ ( if_Code_integer @ ( K4 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_8844_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_8845_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_8846_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_8847_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_8848_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_8849_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_8850_signed__take__bit__add,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_8851_signed__take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_8852_signed__take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_8853_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_8854_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_8855_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X2 ) ) ) ).

% plus_integer.abs_eq
thf(fact_8856_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X2 ) ) ) ).

% minus_integer.abs_eq
thf(fact_8857_times__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa @ X2 ) ) ) ).

% times_integer.abs_eq
thf(fact_8858_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_8859_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_8860_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_8861_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_8862_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_8863_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_8864_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_8865_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_8866_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_8867_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_8868_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_8869_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_8870_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_8871_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_8872_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_8873_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_8874_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_8875_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_8876_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_8877_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_8878_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_8879_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_8880_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_8881_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_8882_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_8883_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_8884_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_8885_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_8886_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_8887_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_8888_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_8889_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_8890_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_8891_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_8892_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_8893_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_8894_polar__Ex,axiom,
    ! [X2: real,Y2: real] :
    ? [R5: real,A3: real] :
      ( ( X2
        = ( times_times_real @ R5 @ ( cos_real @ A3 ) ) )
      & ( Y2
        = ( times_times_real @ R5 @ ( sin_real @ A3 ) ) ) ) ).

% polar_Ex
thf(fact_8895_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_8896_sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( sin_real @ X2 ) @ one_one_real ) ).

% sin_le_one
thf(fact_8897_cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( cos_real @ X2 ) @ one_one_real ) ).

% cos_le_one
thf(fact_8898_log__def,axiom,
    ( log
    = ( ^ [A4: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A4 ) ) ) ) ).

% log_def
thf(fact_8899_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K4: num] : ( suc @ ( pred_numeral @ K4 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_8900_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_8901_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_8902_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_8903_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_8904_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_8905_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_8906_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_8907_sin__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X2 ) ) ).

% sin_ge_minus_one
thf(fact_8908_cos__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X2 ) ) ).

% cos_ge_minus_one
thf(fact_8909_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K4: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K4 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_8910_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_8911_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_8912_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_8913_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_8914_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_8915_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_8916_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_8917_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_8918_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_8919_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_8920_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_8921_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_8922_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_8923_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_8924_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8925_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_8926_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_8927_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_8928_ln__powr__bound,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_8929_ln__powr__bound2,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X2 ) ) ) ) ).

% ln_powr_bound2
thf(fact_8930_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X2 )
             => ( ( log @ A @ X2 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X2 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_8931_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_8932_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_8933_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_8934_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ? [T4: real] :
            ( ( ord_less_real @ X2 @ T4 )
            & ( ord_less_real @ T4 @ zero_zero_real )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M7: nat] : ( times_times_real @ ( cos_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8935_Maclaurin__cos__expansion2,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X2 )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M7: nat] : ( times_times_real @ ( cos_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8936_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X2 )
            & ( ( sin_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M7: nat] : ( times_times_real @ ( sin_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8937_Maclaurin__sin__expansion4,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ? [T4: real] :
          ( ( ord_less_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ X2 )
          & ( ( sin_real @ X2 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M7: nat] : ( times_times_real @ ( sin_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8938_sin__pi__minus,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( minus_minus_real @ pi @ X2 ) )
      = ( sin_real @ X2 ) ) ).

% sin_pi_minus
thf(fact_8939_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_8940_cos__periodic__pi,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( plus_plus_real @ X2 @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).

% cos_periodic_pi
thf(fact_8941_cos__periodic__pi2,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X2 ) )
      = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).

% cos_periodic_pi2
thf(fact_8942_sin__periodic__pi,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( plus_plus_real @ X2 @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).

% sin_periodic_pi
thf(fact_8943_sin__periodic__pi2,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X2 ) )
      = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).

% sin_periodic_pi2
thf(fact_8944_cos__pi__minus,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( minus_minus_real @ pi @ X2 ) )
      = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).

% cos_pi_minus
thf(fact_8945_cos__minus__pi,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( minus_minus_real @ X2 @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).

% cos_minus_pi
thf(fact_8946_sin__minus__pi,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( minus_minus_real @ X2 @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).

% sin_minus_pi
thf(fact_8947_sin__npi2,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8948_sin__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8949_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_8950_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_8951_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8952_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_8953_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_8954_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_8955_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_8956_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_8957_cos__npi2,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% cos_npi2
thf(fact_8958_cos__npi,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% cos_npi
thf(fact_8959_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_8960_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_8961_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_8962_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_8963_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_8964_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_8965_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_8966_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_8967_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_8968_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_8969_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_8970_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_8971_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_8972_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_8973_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_8974_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8975_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8976_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_8977_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_8978_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_8979_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_8980_sin__zero__iff__int2,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( X2
            = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8981_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_8982_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_8983_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_8984_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_8985_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_8986_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_8987_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_8988_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_8989_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_8990_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_8991_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_8992_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_8993_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_8994_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_8995_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_8996_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_8997_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_8998_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_8999_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_9000_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_9001_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_9002_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_9003_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_9004_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_9005_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_9006_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_9007_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_9008_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_9009_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_9010_sincos__total__pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
            & ( ord_less_eq_real @ T4 @ pi )
            & ( X2
              = ( cos_real @ T4 ) )
            & ( Y2
              = ( sin_real @ T4 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_9011_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_9012_cos__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X2
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_9013_sin__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X2
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_9014_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_9015_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_9016_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_9017_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_9018_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_9019_sincos__total__pi__half,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T4: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X2
                = ( cos_real @ T4 ) )
              & ( Y2
                = ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_9020_sincos__total__2pi__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X2
            = ( cos_real @ T4 ) )
          & ( Y2
            = ( sin_real @ T4 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_9021_sincos__total__2pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X2
                  = ( cos_real @ T4 ) )
               => ( Y2
                 != ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_9022_Maclaurin__sin__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( sin_real @ X2 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M7: nat] : ( times_times_real @ ( sin_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_9023_Maclaurin__cos__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
      & ( ( cos_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M7: nat] : ( times_times_real @ ( cos_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_9024_Maclaurin__sin__expansion2,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
      & ( ( sin_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M7: nat] : ( times_times_real @ ( sin_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_9025_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_9026_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_9027_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_9028_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_9029_abs__sin__x__le__abs__x,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ ( abs_abs_real @ X2 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_9030_lemma__interval__lt,axiom,
    ! [A: real,X2: real,B: real] :
      ( ( ord_less_real @ A @ X2 )
     => ( ( ord_less_real @ X2 @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y4 )
                  & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_9031_abs__cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X2 ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_9032_abs__sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_9033_sin__bound__lemma,axiom,
    ! [X2: real,Y2: real,U: real,V: real] :
      ( ( X2 = Y2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y2 ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_9034_lemma__interval,axiom,
    ! [A: real,X2: real,B: real] :
      ( ( ord_less_real @ A @ X2 )
     => ( ( ord_less_real @ X2 @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_9035_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_9036_sin__cos__le1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_9037_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_9038_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_9039_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K4: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_9040_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_9041_arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( arctan @ X2 )
        = ( suminf_real
          @ ^ [K4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K4 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_9042_summable__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( summable_real
        @ ^ [K4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K4 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_9043_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_9044_zdvd1__eq,axiom,
    ! [X2: int] :
      ( ( dvd_dvd_int @ X2 @ one_one_int )
      = ( ( abs_abs_int @ X2 )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_9045_tanh__real__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% tanh_real_le_iff
thf(fact_9046_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_9047_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_9048_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_9049_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_9050_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_9051_summable__rabs__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
     => ( summable_real @ F ) ) ).

% summable_rabs_cancel
thf(fact_9052_arctan__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% arctan_le_iff
thf(fact_9053_arctan__monotone_H,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) ) ) ).

% arctan_monotone'
thf(fact_9054_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_9055_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_9056_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M7: int] :
          ? [N: int] :
            ( ( ord_less_eq_int @ M7 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_9057_tanh__real__lt__1,axiom,
    ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_9058_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_9059_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_9060_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_9061_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N10: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N10 @ 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_9062_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_9063_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_9064_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9065_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_9066_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_9067_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_abs_add_iff
thf(fact_9068_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_add_abs_iff
thf(fact_9069_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9070_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_9071_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_9072_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M @ I3 )
            & ( ord_less_nat @ I3 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N2 )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_9073_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_9074_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_9075_arctan__ubound,axiom,
    ! [Y2: real] : ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_9076_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_9077_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_9078_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_9079_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_9080_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_9081_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_9082_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_9083_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_9084_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_9085_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_9086_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_9087_tan__periodic__pi,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ pi ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_pi
thf(fact_9088_tan__npi,axiom,
    ! [N2: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_9089_tan__periodic__n,axiom,
    ! [X2: real,N2: num] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ N2 ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_n
thf(fact_9090_tan__periodic__nat,axiom,
    ! [X2: real,N2: nat] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_nat
thf(fact_9091_tan__periodic__int,axiom,
    ! [X2: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_int
thf(fact_9092_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_9093_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_9094_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_9095_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_9096_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_9097_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_9098_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_9099_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_9100_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_9101_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_9102_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_9103_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_9104_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_9105_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_9106_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_9107_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_9108_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_9109_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_9110_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_9111_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_9112_tan__total__pi4,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ? [Z3: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
          & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z3 )
            = X2 ) ) ) ).

% tan_total_pi4
thf(fact_9113_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T4 ) @ ( sin_real @ T4 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_9114_Maclaurin__exp__lt,axiom,
    ! [X2: real,N2: nat] :
      ( ( X2 != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
            & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
            & ( ( exp_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M7: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M7 ) @ ( semiri2265585572941072030t_real @ M7 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_9115_exp__le__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% exp_le_cancel_iff
thf(fact_9116_exp__eq__one__iff,axiom,
    ! [X2: real] :
      ( ( ( exp_real @ X2 )
        = one_one_real )
      = ( X2 = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_9117_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_9118_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_9119_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_9120_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_9121_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_9122_Complex__mult__complex__of__real,axiom,
    ! [X2: real,Y2: real,R3: real] :
      ( ( times_times_complex @ ( complex2 @ X2 @ Y2 ) @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ ( times_times_real @ X2 @ R3 ) @ ( times_times_real @ Y2 @ R3 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_9123_complex__of__real__mult__Complex,axiom,
    ! [R3: real,X2: real,Y2: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( complex2 @ X2 @ Y2 ) )
      = ( complex2 @ ( times_times_real @ R3 @ X2 ) @ ( times_times_real @ R3 @ Y2 ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_9124_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% complex_diff
thf(fact_9125_complex__scaleR,axiom,
    ! [R3: real,A: real,B: real] :
      ( ( real_V2046097035970521341omplex @ R3 @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( times_times_real @ R3 @ A ) @ ( times_times_real @ R3 @ B ) ) ) ).

% complex_scaleR
thf(fact_9126_exp__ge__zero,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).

% exp_ge_zero
thf(fact_9127_not__exp__le__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_9128_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_9129_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% complex_add
thf(fact_9130_Complex__add__complex__of__real,axiom,
    ! [X2: real,Y2: real,R3: real] :
      ( ( plus_plus_complex @ ( complex2 @ X2 @ Y2 ) @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ ( plus_plus_real @ X2 @ R3 ) @ Y2 ) ) ).

% Complex_add_complex_of_real
thf(fact_9131_complex__of__real__add__Complex,axiom,
    ! [R3: real,X2: real,Y2: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( complex2 @ X2 @ Y2 ) )
      = ( complex2 @ ( plus_plus_real @ R3 @ X2 ) @ Y2 ) ) ).

% complex_of_real_add_Complex
thf(fact_9132_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_9133_exp__ge__add__one__self,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ).

% exp_ge_add_one_self
thf(fact_9134_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_9135_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_9136_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% complex_mult
thf(fact_9137_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_9138_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_9139_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_9140_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_9141_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_9142_ln__x__over__x__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y2 ) @ Y2 ) @ ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_9143_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_9144_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_9145_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_9146_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_9147_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_9148_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_9149_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_9150_Maclaurin__exp__le,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
      & ( ( exp_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M7: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M7 ) @ ( semiri2265585572941072030t_real @ M7 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9151_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_9152_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_9153_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_9154_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_9155_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_9156_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_9157_real__sqrt__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ X2 )
        = ( sqrt @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% real_sqrt_eq_iff
thf(fact_9158_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_9159_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_9160_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_9161_real__sqrt__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% real_sqrt_le_iff
thf(fact_9162_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_9163_real__sqrt__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( ( sqrt @ X2 )
        = one_one_real )
      = ( X2 = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_9164_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_9165_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_9166_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_9167_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_9168_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_9169_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_9170_real__sqrt__le__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ one_one_real )
      = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_9171_real__sqrt__ge__1__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y2 ) )
      = ( ord_less_eq_real @ one_one_real @ Y2 ) ) ).

% real_sqrt_ge_1_iff
thf(fact_9172_real__sqrt__abs2,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( times_times_real @ X2 @ X2 ) )
      = ( abs_abs_real @ X2 ) ) ).

% real_sqrt_abs2
thf(fact_9173_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_9174_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_9175_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_9176_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_9177_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_9178_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_9179_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A3: complex,R5: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( exp_complex @ A3 ) ) ) ).

% complex_exp_exists
thf(fact_9180_real__sqrt__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_le_mono
thf(fact_9181_real__sqrt__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( sqrt @ ( times_times_real @ X2 @ Y2 ) )
      = ( times_times_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_mult
thf(fact_9182_real__sqrt__divide,axiom,
    ! [X2: real,Y2: real] :
      ( ( sqrt @ ( divide_divide_real @ X2 @ Y2 ) )
      = ( divide_divide_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_divide
thf(fact_9183_real__sqrt__power,axiom,
    ! [X2: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X2 @ K ) )
      = ( power_power_real @ ( sqrt @ X2 ) @ K ) ) ).

% real_sqrt_power
thf(fact_9184_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_9185_real__sqrt__minus,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X2 ) )
      = ( uminus_uminus_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_minus
thf(fact_9186_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_9187_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_9188_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_9189_real__sqrt__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_one
thf(fact_9190_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_9191_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_9192_le__real__sqrt__sumsq,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_9193_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_9194_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_9195_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_9196_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_9197_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_9198_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_9199_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_9200_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_9201_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_9202_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_9203_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_9204_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_9205_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_9206_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_9207_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_9208_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_9209_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_9210_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_9211_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_9212_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_9213_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_9214_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_9215_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_9216_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_9217_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_9218_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_9219_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_9220_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_9221_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_9222_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_9223_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_9224_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_9225_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_9226_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_9227_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_9228_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_9229_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_9230_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_9231_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_9232_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_9233_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_9234_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_9235_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_9236_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_9237_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_9238_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_9239_cos__arccos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y2 ) )
          = Y2 ) ) ) ).

% cos_arccos
thf(fact_9240_sin__arcsin,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y2 ) )
          = Y2 ) ) ) ).

% sin_arcsin
thf(fact_9241_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_9242_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_9243_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_9244_arccos__le__arccos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_9245_arccos__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real ) )
     => ( ( ( arccos @ X2 )
          = ( arccos @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% arccos_eq_iff
thf(fact_9246_arccos__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
          = ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).

% arccos_le_mono
thf(fact_9247_arcsin__le__arcsin,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_9248_arcsin__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X2 ) )
          = ( uminus_uminus_real @ ( arcsin @ X2 ) ) ) ) ) ).

% arcsin_minus
thf(fact_9249_arcsin__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ( arcsin @ X2 )
            = ( arcsin @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% arcsin_eq_iff
thf(fact_9250_arcsin__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
          = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).

% arcsin_le_mono
thf(fact_9251_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_9252_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_9253_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_9254_arccos__ubound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_9255_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_9256_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_9257_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_9258_cos__arccos__abs,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y2 ) )
        = Y2 ) ) ).

% cos_arccos_abs
thf(fact_9259_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_9260_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_9261_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_9262_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_9263_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_9264_arccos__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
          = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_minus
thf(fact_9265_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_9266_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_9267_arccos__minus__abs,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
        = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ).

% arccos_minus_abs
thf(fact_9268_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_9269_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_9270_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_9271_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_9272_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_9273_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_9274_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_9275_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_9276_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_9277_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_9278_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_9279_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_9280_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_9281_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_9282_complex__i__mult__minus,axiom,
    ! [X2: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X2 ) )
      = ( uminus1482373934393186551omplex @ X2 ) ) ).

% complex_i_mult_minus
thf(fact_9283_divide__i,axiom,
    ! [X2: complex] :
      ( ( divide1717551699836669952omplex @ X2 @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X2 ) ) ).

% divide_i
thf(fact_9284_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_9285_divide__numeral__i,axiom,
    ! [Z: complex,N2: num] :
      ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N2 ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% divide_numeral_i
thf(fact_9286_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9287_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_9288_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_9289_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_9290_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
thf(fact_9291_complex__i__not__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( numera6690914467698888265omplex @ W ) ) ).

% complex_i_not_numeral
thf(fact_9292_real__sqrt__inverse,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X2 ) )
      = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_inverse
thf(fact_9293_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).

% divide_real_def
thf(fact_9294_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W )
        = Z )
      = ( W
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).

% i_times_eq_iff
thf(fact_9295_complex__i__not__neg__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% complex_i_not_neg_numeral
thf(fact_9296_inverse__powr,axiom,
    ! [Y2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( powr_real @ ( inverse_inverse_real @ Y2 ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y2 @ A ) ) ) ) ).

% inverse_powr
thf(fact_9297_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_9298_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9299_i__mult__Complex,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% i_mult_Complex
thf(fact_9300_Complex__mult__i,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% Complex_mult_i
thf(fact_9301_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9302_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_9303_i__complex__of__real,axiom,
    ! [R3: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ zero_zero_real @ R3 ) ) ).

% i_complex_of_real
thf(fact_9304_complex__of__real__i,axiom,
    ! [R3: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R3 ) ) ).

% complex_of_real_i
thf(fact_9305_log__inverse,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( log @ A @ ( inverse_inverse_real @ X2 ) )
            = ( uminus_uminus_real @ ( log @ A @ X2 ) ) ) ) ) ) ).

% log_inverse
thf(fact_9306_Complex__eq,axiom,
    ( complex2
    = ( ^ [A4: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A4 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).

% Complex_eq
thf(fact_9307_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_9308_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R5: real,A3: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_9309_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_9310_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_9311_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_9312_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_9313_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_9314_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_9315_cmod__complex__polar,axiom,
    ! [R3: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R3 ) ) ).

% cmod_complex_polar
thf(fact_9316_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_9317_Maclaurin__sin__bound,axiom,
    ! [X2: real,N2: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X2 )
          @ ( groups6591440286371151544t_real
            @ ^ [M7: nat] : ( times_times_real @ ( sin_coeff @ M7 ) @ ( power_power_real @ X2 @ M7 ) )
            @ ( 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_9318_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_9319_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_9320_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9321_csqrt__1,axiom,
    ( ( csqrt @ one_one_complex )
    = one_one_complex ) ).

% csqrt_1
thf(fact_9322_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% csqrt_eq_1
thf(fact_9323_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9324_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).

% divide_complex_def
thf(fact_9325_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_9326_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_9327_complex__inverse,axiom,
    ! [A: real,B: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_9328_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_9329_sinh__real__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% sinh_real_le_iff
thf(fact_9330_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_9331_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_9332_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_9333_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9334_cis__pi,axiom,
    ( ( cis @ pi )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cis_pi
thf(fact_9335_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9336_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9337_sinh__le__cosh__real,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).

% sinh_le_cosh_real
thf(fact_9338_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_9339_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_9340_cosh__real__nonneg,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_nonneg
thf(fact_9341_cosh__real__ge__1,axiom,
    ! [X2: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_ge_1
thf(fact_9342_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_9343_cis__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).

% cis_divide
thf(fact_9344_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_9345_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_9346_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_9347_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_9348_DeMoivre,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N2 )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) ) ) ).

% DeMoivre
thf(fact_9349_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B4: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_9350_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_9351_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_9352_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( bij_betw_nat_complex
        @ ^ [K4: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K4 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9353_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_9354_cot__npi,axiom,
    ! [N2: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9355_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_9356_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_9357_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_9358_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_9359_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_9360_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_9361_semiring__norm_I28_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ N2 ) ) ) ).

% semiring_norm(28)
thf(fact_9362_semiring__norm_I27_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit0 @ N2 ) )
      = ( bit1 @ ( bitM @ N2 ) ) ) ).

% semiring_norm(27)
thf(fact_9363_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_9364_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_9365_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_9366_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
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9367_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : ( entails @ ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ).

% assnle
thf(fact_9368_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z2: int] :
                ( ( ord_less_eq_int @ D2 @ Z2 )
                & ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9369_real__root__zero,axiom,
    ! [N2: nat] :
      ( ( root @ N2 @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9370_real__root__Suc__0,axiom,
    ! [X2: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
      = X2 ) ).

% real_root_Suc_0
thf(fact_9371_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_9372_root__0,axiom,
    ! [X2: real] :
      ( ( root @ zero_zero_nat @ X2 )
      = zero_zero_real ) ).

% root_0
thf(fact_9373_ent__pure__pre__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ Q )
      = ( B
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_9374_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q )
      = ( B
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_9375_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_9376_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_9377_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_9378_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_9379_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_9380_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( ( entails @ P @ ( times_times_assn @ Q @ ( pure_assn @ B ) ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_9381_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: $o] :
      ( ( entails @ P @ ( pure_assn @ B ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_9382_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_9383_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_9384_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_9385_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_9386_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_9387_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_9388_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_9389_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_9390_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_9391_real__root__inverse,axiom,
    ! [N2: nat,X2: real] :
      ( ( root @ N2 @ ( inverse_inverse_real @ X2 ) )
      = ( inverse_inverse_real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_inverse
thf(fact_9392_fr__rot,axiom,
    ! [A2: assn,B5: assn,C4: assn] :
      ( ( entails @ ( times_times_assn @ A2 @ B5 ) @ C4 )
     => ( entails @ ( times_times_assn @ B5 @ A2 ) @ C4 ) ) ).

% fr_rot
thf(fact_9393_fr__refl,axiom,
    ! [A2: assn,B5: assn,C4: assn] :
      ( ( entails @ A2 @ B5 )
     => ( entails @ ( times_times_assn @ A2 @ C4 ) @ ( times_times_assn @ B5 @ C4 ) ) ) ).

% fr_refl
thf(fact_9394_fr__rot__rhs,axiom,
    ! [A2: assn,B5: assn,C4: assn] :
      ( ( entails @ A2 @ ( times_times_assn @ B5 @ C4 ) )
     => ( entails @ A2 @ ( times_times_assn @ C4 @ B5 ) ) ) ).

% fr_rot_rhs
thf(fact_9395_ent__frame__fwd,axiom,
    ! [P: assn,R2: assn,Ps: assn,F5: assn,Q: assn] :
      ( ( entails @ P @ R2 )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F5 ) )
       => ( ( entails @ ( times_times_assn @ R2 @ F5 ) @ Q )
         => ( entails @ Ps @ Q ) ) ) ) ).

% ent_frame_fwd
thf(fact_9396_ent__star__mono,axiom,
    ! [P: assn,P6: assn,Q: assn,Q7: assn] :
      ( ( entails @ P @ P6 )
     => ( ( entails @ Q @ Q7 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P6 @ Q7 ) ) ) ) ).

% ent_star_mono
thf(fact_9397_real__root__mult,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( root @ N2 @ ( times_times_real @ X2 @ Y2 ) )
      = ( times_times_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ).

% real_root_mult
thf(fact_9398_real__root__mult__exp,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( root @ ( times_times_nat @ M @ N2 ) @ X2 )
      = ( root @ M @ ( root @ N2 @ X2 ) ) ) ).

% real_root_mult_exp
thf(fact_9399_real__root__minus,axiom,
    ! [N2: nat,X2: real] :
      ( ( root @ N2 @ ( uminus_uminus_real @ X2 ) )
      = ( uminus_uminus_real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_minus
thf(fact_9400_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ Q ) ) ).

% is_entails
thf(fact_9401_real__root__commute,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( root @ M @ ( root @ N2 @ X2 ) )
      = ( root @ N2 @ ( root @ M @ X2 ) ) ) ).

% real_root_commute
thf(fact_9402_real__root__divide,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( root @ N2 @ ( divide_divide_real @ X2 @ Y2 ) )
      = ( divide_divide_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ).

% real_root_divide
thf(fact_9403_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_9404_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: produc3658429121746597890et_nat,P: assn,R2: assn,F5: assn] :
      ( ( rep_assn @ Ps @ H2 )
     => ( ( entails @ P @ R2 )
       => ( ( entails @ Ps @ ( times_times_assn @ P @ F5 ) )
         => ( rep_assn @ ( times_times_assn @ R2 @ F5 ) @ H2 ) ) ) ) ).

% mod_frame_fwd
thf(fact_9405_ent__star__mono__true,axiom,
    ! [A2: assn,A9: assn,B5: assn,B10: assn] :
      ( ( entails @ A2 @ ( times_times_assn @ A9 @ top_top_assn ) )
     => ( ( entails @ B5 @ ( times_times_assn @ B10 @ top_top_assn ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ A2 @ B5 ) @ top_top_assn ) @ ( times_times_assn @ ( times_times_assn @ A9 @ B10 ) @ top_top_assn ) ) ) ) ).

% ent_star_mono_true
thf(fact_9406_ent__refl__true,axiom,
    ! [A2: assn] : ( entails @ A2 @ ( times_times_assn @ A2 @ top_top_assn ) ) ).

% ent_refl_true
thf(fact_9407_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R2: assn] :
      ( ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) )
     => ( entails @ ( times_times_assn @ P @ R2 ) @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(1)
thf(fact_9408_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(2)
thf(fact_9409_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_9410_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_9411_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_9412_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_9413_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_9414_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_9415_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9416_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_9417_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_9418_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_9419_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_9420_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_9421_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_9422_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_9423_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_9424_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_9425_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_9426_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_9427_log__root,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N2 @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_9428_log__base__root,axiom,
    ! [N2: nat,B: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N2 @ B ) @ X2 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X2 ) ) ) ) ) ).

% log_base_root
thf(fact_9429_ln__root,axiom,
    ! [N2: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N2 @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_9430_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_9431_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z2: int] :
                ( ( ord_less_eq_int @ D2 @ Z7 )
                & ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9432_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,L: nat,X2: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails
        @ ( times_times_assn
          @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) @ X2 )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ 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 @ Summary @ X14 ) )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
          @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) ) ) ) ).

% big_assn_simp
thf(fact_9433_local_Oext,axiom,
    ! [Y2: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y2 ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ Y2 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y2 ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ Y2 ) ) ) ) ) ).

% local.ext
thf(fact_9434_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9435_txe,axiom,
    ! [Y2: nat,TreeList: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y2 ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ Y2 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) ).

% txe
thf(fact_9436_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ 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 @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) ).

% repack
thf(fact_9437_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ I ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ I ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) ).

% recomp
thf(fact_9438_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X2: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( Xaa
          = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
       => ( 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 @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ 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 @ Summary @ X14 ) )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_9439_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M7: nat] :
            ( ( ord_less_nat @ M7 @ N2 )
            & ( P @ M7 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9440_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M7: nat] :
            ( ( ord_less_nat @ M7 @ N2 )
           => ( P @ M7 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9441_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9442_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_9443_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_9444_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_9445_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_9446_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_9447_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_9448_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_9449_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_9450_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_9451_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I3 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9452_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_9453_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
      = ( set_or189985376899183464nteger @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_9454_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9455_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_9456_forall__finite_I3_J,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ ( suc @ X2 ) ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ X2 ) )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_9457_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_9458_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_9459_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_9460_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R3: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R3 ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_9461_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_9462_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q5: int,R: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_9463_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_9464_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I4: nat] :
      ( ( ord_less_nat @ I4 @ zero_zero_nat )
     => ( P @ I4 ) ) ).

% forall_finite(1)
thf(fact_9465_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_9466_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_9467_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ X2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ X2 )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_9468_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ zero_zero_nat ) )
           => ( P @ I2 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_9469_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9470_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_9471_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9472_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9473_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_9474_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_9475_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_9476_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_9477_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_9478_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_9479_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_9480_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_9481_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_9482_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_9483_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_9484_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_9485_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_9486_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9487_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L2: int] :
          ( if_int
          @ ( ( K4 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K4
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K4
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9488_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_9489_uint32_Osize__eq,axiom,
    ( size_size_uint32
    = ( ^ [P5: uint32] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_9490_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_9491_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_9492_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_9493_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_9494_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_9495_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_9496_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M7: nat,N: nat] :
          ( if_nat
          @ ( ( M7 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M7 @ ( 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 @ M7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9497_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M7: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M7 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9498_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_9499_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_9500_take__bit__num__simps_I5_J,axiom,
    ! [R3: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9501_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9502_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9503_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_9504_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_9505_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9506_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9507_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
        @ ^ [Q5: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q5 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9508_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_9509_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_9510_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_9511_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_9512_take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_9513_take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).

% take_bit_diff
thf(fact_9514_add__inc,axiom,
    ! [X2: num,Y2: num] :
      ( ( plus_plus_num @ X2 @ ( inc @ Y2 ) )
      = ( inc @ ( plus_plus_num @ X2 @ Y2 ) ) ) ).

% add_inc
thf(fact_9515_num__induct,axiom,
    ! [P: num > $o,X2: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X2 ) ) ) ).

% num_induct
thf(fact_9516_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_9517_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_9518_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_9519_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9520_inc_Osimps_I2_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit0 @ X2 ) )
      = ( bit1 @ X2 ) ) ).

% inc.simps(2)
thf(fact_9521_inc_Osimps_I3_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit1 @ X2 ) )
      = ( bit0 @ ( inc @ X2 ) ) ) ).

% inc.simps(3)
thf(fact_9522_add__One,axiom,
    ! [X2: num] :
      ( ( plus_plus_num @ X2 @ one )
      = ( inc @ X2 ) ) ).

% add_One
thf(fact_9523_inc__BitM__eq,axiom,
    ! [N2: num] :
      ( ( inc @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% inc_BitM_eq
thf(fact_9524_BitM__inc__eq,axiom,
    ! [N2: num] :
      ( ( bitM @ ( inc @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% BitM_inc_eq
thf(fact_9525_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_9526_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_9527_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_9528_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_9529_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_9530_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,M7: nat] : ( modulo_modulo_nat @ M7 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_nat_def
thf(fact_9531_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_9532_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_9533_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,K4: int] : ( modulo_modulo_int @ K4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_def
thf(fact_9534_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_9535_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_9536_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_9537_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_9538_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_9539_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_9540_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_9541_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_9542_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_9543_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_9544_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_9545_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K4: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K4 @ ( 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_9546_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_9547_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_9548_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_9549_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_9550_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_9551_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_9552_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_9553_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_9554_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9555_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_9556_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_9557_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_9558_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_9559_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_9560_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_9561_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_9562_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_9563_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_9564_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_9565_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_9566_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 )
        @ ^ [N11: num] : ( some_num @ ( bit1 @ N11 ) )
        @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_9567_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_9568_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_9569_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_9570_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K4: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K4
          @ ( if_int
            @ ( ( sgn_sgn_int @ K4 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L2 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L2 @ K4 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9571_sgn__mult__dvd__iff,axiom,
    ! [R3: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R3 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9572_mult__sgn__dvd__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R3 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R3 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9573_dvd__sgn__mult__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9574_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R3: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R3 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9575_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L ) )
     => ( ( divide_divide_int @ K @ L )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_9576_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N4: nat,L3: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% int_sgnE
thf(fact_9577_ln__real__def,axiom,
    ( ln_ln_real
    = ( ^ [X: real] :
          ( the_real
          @ ^ [U2: real] :
              ( ( exp_real @ U2 )
              = X ) ) ) ) ).

% ln_real_def
thf(fact_9578_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_9579_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_9580_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9581_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_9582_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( dvd_dvd_int @ L @ K )
     => ( ( divide_divide_int @ K @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_9583_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_9584_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R3 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R3 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R3 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9585_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9586_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A32 )
     => ( ( ( A23 = zero_zero_int )
         => ( A32
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q3: int] :
              ( ( A32
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q3 @ A23 ) ) ) )
         => ~ ! [R5: int,Q3: int] :
                ( ( A32
                  = ( product_Pair_int_int @ Q3 @ R5 ) )
               => ( ( ( sgn_sgn_int @ R5 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R5 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9587_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
          ( ? [K4: int] :
              ( ( A1 = K4 )
              & ( A22 = zero_zero_int )
              & ( A33
                = ( product_Pair_int_int @ zero_zero_int @ K4 ) ) )
          | ? [L2: int,K4: int,Q5: int] :
              ( ( A1 = K4 )
              & ( A22 = L2 )
              & ( A33
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K4
                = ( times_times_int @ Q5 @ L2 ) ) )
          | ? [R: int,L2: int,K4: int,Q5: int] :
              ( ( A1 = K4 )
              & ( A22 = L2 )
              & ( A33
                = ( product_Pair_int_int @ Q5 @ R ) )
              & ( ( sgn_sgn_int @ R )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L2 ) )
              & ( K4
                = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9588_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_9589_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_9590_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_9591_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
            @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
            @ ( bit_take_bit_num @ N @ M ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9592_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9593_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N2
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9594_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K4: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K4 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K4 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9595_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_9596_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M7: num] :
          ( produc478579273971653890on_num
          @ ^ [A4: 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
                      @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X )
              @ A4 )
          @ ( product_Pair_nat_num @ N @ M7 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9597_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( divide_divide_int @ A @ B )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( divide_divide_int @ A @ B ) )
        = ( sgn_sgn_int @ ( times_times_int @ A @ B ) ) ) ) ).

% sgn_div_eq_sgn_mult
thf(fact_9598_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_9599_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_9600_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu3: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_9601_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu3: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_9602_real__sgn__eq,axiom,
    ( sgn_sgn_real
    = ( ^ [X: real] : ( divide_divide_real @ X @ ( abs_abs_real @ X ) ) ) ) ).

% real_sgn_eq
thf(fact_9603_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_9604_sgn__eq,axiom,
    ( sgn_sgn_complex
    = ( ^ [Z2: complex] : ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).

% sgn_eq
thf(fact_9605_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_9606_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_9607_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_9608_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9609_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K4: code_integer] : ( if_Code_integer @ ( K4 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9610_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K4: nat] : K4
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_9611_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X2: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X2 )
     => ( ( X2
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9612_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_9613_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_9614_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_9615_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_9616_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9617_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_9618_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_9619_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9620_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_9621_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9622_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R3 )
     => ~ ! [S: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S )
           => ! [T4: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T4 )
               => ( R3
                 != ( plus_plus_rat @ S @ T4 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9623_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_9624_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9625_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9626_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9627_Arg__def,axiom,
    ( arg
    = ( ^ [Z2: complex] :
          ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A4: real] :
                ( ( ( sgn_sgn_complex @ Z2 )
                  = ( cis @ A4 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
                & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9628_new__addr__refl,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( hoare_new_addrs @ H2 @ As3 @ H2 )
      = As3 ) ).

% new_addr_refl
thf(fact_9629_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q5: rat,R: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R ) ) ) ) ).

% diff_rat_def
thf(fact_9630_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K4: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K4 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K4 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9631_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_9632_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_9633_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_9634_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_9635_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_9636_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_9637_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_9638_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_9639_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_9640_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_9641_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K4: int] : ( minus_minus_int @ ( uminus_uminus_int @ K4 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9642_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_9643_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_9644_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_9645_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_9646_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_9647_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_9648_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_9649_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_9650_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ N4 @ M3 )
             => ( ( bit_se1146084159140164899it_int @ K @ M3 )
                = ( 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_9651_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_9652_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_9653_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_9654_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_9655_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_9656_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_9657_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_9658_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_9659_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K4: int,N: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_int_def
thf(fact_9660_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_9661_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K4: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9662_Bit__Operations_Oset__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K4: int] :
          ( plus_plus_int @ K4
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K4 @ N ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% Bit_Operations.set_bit_eq
thf(fact_9663_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K4: int] : ( minus_minus_int @ K4 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K4 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9664_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_9665_int__not__code_I1_J,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_not_code(1)
thf(fact_9666_bitNOT__integer__code,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [I2: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ I2 ) @ one_one_Code_integer ) ) ) ).

% bitNOT_integer_code
thf(fact_9667_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9668_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_9669_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_9670_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_9671_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_9672_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_9673_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_9674_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_9675_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_9676_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9677_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_9678_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_9679_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_9680_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_9681_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_9682_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_9683_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_9684_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_9685_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_9686_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_9687_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_9688_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M7: nat,N: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M7 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_nat_def
thf(fact_9689_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_9690_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_9691_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_9692_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_9693_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_9694_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M7: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M7 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9695_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_9696_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_9697_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K4: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9698_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M7: nat,N: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M7 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M7 @ ( 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 @ M7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9699_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K4: int,L2: int] :
          ( if_int
          @ ( ( K4
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K4 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K4 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9700_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_9701_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K4: int,L2: int] :
          ( if_int
          @ ( K4
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K4 )
            @ ( if_int @ ( K4 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K4 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9702_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9703_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_9704_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_9705_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K4: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9706_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_9707_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_9708_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_9709_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_9710_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_9711_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_9712_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_9713_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_9714_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_9715_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_9716_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9717_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_9718_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_9719_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_9720_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_9721_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_9722_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_9723_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_9724_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 )
           => ! [M2: num] :
                ( ( Xa
                  = ( bit0 @ M2 ) )
               => ( Y2
                 != ( bit1 @ M2 ) ) ) )
         => ( ( ( X2 = one )
             => ! [M2: num] :
                  ( ( Xa
                    = ( bit1 @ M2 ) )
                 => ( Y2
                   != ( bit1 @ M2 ) ) ) )
           => ( ( ? [N4: num] :
                    ( X2
                    = ( bit0 @ N4 ) )
               => ( ( Xa = one )
                 => ( Y2
                   != ( bit0 @ one ) ) ) )
             => ( ! [N4: num] :
                    ( ( X2
                      = ( bit0 @ N4 ) )
                   => ! [M2: num] :
                        ( ( Xa
                          = ( bit0 @ M2 ) )
                       => ( Y2
                         != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X2
                        = ( bit0 @ N4 ) )
                     => ! [M2: num] :
                          ( ( Xa
                            = ( bit1 @ M2 ) )
                         => ( Y2
                           != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
                 => ( ( ? [N4: num] :
                          ( X2
                          = ( bit1 @ N4 ) )
                     => ( ( Xa = one )
                       => ( Y2 != one ) ) )
                   => ( ! [N4: num] :
                          ( ( X2
                            = ( bit1 @ N4 ) )
                         => ! [M2: num] :
                              ( ( Xa
                                = ( bit0 @ M2 ) )
                             => ( Y2
                               != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
                     => ~ ! [N4: num] :
                            ( ( X2
                              = ( bit1 @ N4 ) )
                           => ! [M2: num] :
                                ( ( Xa
                                  = ( bit1 @ M2 ) )
                               => ( Y2
                                 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9725_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_9726_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_9727_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_9728_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M7: nat,N: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M7 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M7 @ ( 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 @ M7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9729_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M7: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M7 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9730_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_9731_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_9732_or__not__num__neg_Opelims,axiom,
    ! [X2: num,Xa: num,Y2: num] :
      ( ( ( bit_or_not_num_neg @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X2 @ Xa ) )
       => ( ( ( X2 = one )
           => ( ( Xa = one )
             => ( ( Y2 = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X2 = one )
             => ! [M2: num] :
                  ( ( Xa
                    = ( bit0 @ M2 ) )
                 => ( ( Y2
                      = ( bit1 @ M2 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M2 ) ) ) ) ) )
           => ( ( ( X2 = one )
               => ! [M2: num] :
                    ( ( Xa
                      = ( bit1 @ M2 ) )
                   => ( ( Y2
                        = ( bit1 @ M2 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M2 ) ) ) ) ) )
             => ( ! [N4: num] :
                    ( ( X2
                      = ( bit0 @ N4 ) )
                   => ( ( Xa = one )
                     => ( ( Y2
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ one ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X2
                        = ( bit0 @ N4 ) )
                     => ! [M2: num] :
                          ( ( Xa
                            = ( bit0 @ M2 ) )
                         => ( ( Y2
                              = ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit0 @ M2 ) ) ) ) ) )
                 => ( ! [N4: num] :
                        ( ( X2
                          = ( bit0 @ N4 ) )
                       => ! [M2: num] :
                            ( ( Xa
                              = ( bit1 @ M2 ) )
                           => ( ( Y2
                                = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit1 @ M2 ) ) ) ) ) )
                   => ( ! [N4: num] :
                          ( ( X2
                            = ( bit1 @ N4 ) )
                         => ( ( Xa = one )
                           => ( ( Y2 = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ one ) ) ) ) )
                     => ( ! [N4: num] :
                            ( ( X2
                              = ( bit1 @ N4 ) )
                           => ! [M2: num] :
                                ( ( Xa
                                  = ( bit0 @ M2 ) )
                               => ( ( Y2
                                    = ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit0 @ M2 ) ) ) ) ) )
                       => ~ ! [N4: num] :
                              ( ( X2
                                = ( bit1 @ N4 ) )
                             => ! [M2: num] :
                                  ( ( Xa
                                    = ( bit1 @ M2 ) )
                                 => ( ( Y2
                                      = ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit1 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_9733_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_9734_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9735_num_Osize__gen_I2_J,axiom,
    ! [X23: num] :
      ( ( size_num @ ( bit0 @ X23 ) )
      = ( plus_plus_nat @ ( size_num @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_9736_dup__1,axiom,
    ( ( code_dup @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dup_1
thf(fact_9737_dup_Oabs__eq,axiom,
    ! [X2: int] :
      ( ( code_dup @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( plus_plus_int @ X2 @ X2 ) ) ) ).

% dup.abs_eq
thf(fact_9738_pow_Osimps_I1_J,axiom,
    ! [X2: num] :
      ( ( pow @ X2 @ one )
      = X2 ) ).

% pow.simps(1)
thf(fact_9739_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_9740_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_9741_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_9742_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_9743_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B4 ) @ ( abs_abs_int @ A4 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_9744_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_9745_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9746_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9747_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_9748_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_9749_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q5: rat,R: rat] : ( times_times_rat @ Q5 @ ( inverse_inverse_rat @ R ) ) ) ) ).

% divide_rat_def
thf(fact_9750_quotient__of__div,axiom,
    ! [R3: rat,N2: int,D: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ N2 @ D ) )
     => ( R3
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_9751_quotient__of__denom__pos,axiom,
    ! [R3: rat,P4: int,Q2: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_9752_quotient__of__denom__pos_H,axiom,
    ! [R3: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R3 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9753_rat__abs__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int] : ( product_Pair_int_int @ ( abs_abs_int @ A4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_abs_code
thf(fact_9754_rat__uminus__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_uminus_code
thf(fact_9755_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q5: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D2: int] : ( ord_less_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9756_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P5: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P5 ) ) ) ) ).

% rat_floor_code
thf(fact_9757_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q5: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9758_rat__sgn__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9759_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9760_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_9761_rat__minus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_minus_code
thf(fact_9762_normalize__denom__zero,axiom,
    ! [P4: int] :
      ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9763_normalize__negative,axiom,
    ! [Q2: int,P4: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_9764_normalize__denom__pos,axiom,
    ! [R3: product_prod_int_int,P4: int,Q2: int] :
      ( ( ( normalize @ R3 )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_9765_normalize__crossproduct,axiom,
    ! [Q2: int,S3: int,P4: int,R3: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S3 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R3 @ S3 ) ) )
         => ( ( times_times_int @ P4 @ S3 )
            = ( times_times_int @ R3 @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9766_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9767_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9768_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_9769_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_9770_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_9771_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_9772_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_9773_rat__times__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B4 ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_times_code
thf(fact_9774_rat__divide__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C3 @ B4 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_9775_rat__plus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_9776_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_9777_setceilmax,axiom,
    ! [S3: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ S3 @ 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 @ S3 ) )
                = ( 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 @ S3 @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_9778_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% height_compose_list
thf(fact_9779_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd_int @ M @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_9780_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_9781_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_9782_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_9783_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_9784_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_9785_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_9786_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_9787_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N2 ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9788_bezout__int,axiom,
    ! [X2: int,Y2: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X2 ) @ ( times_times_int @ V2 @ Y2 ) )
      = ( gcd_gcd_int @ X2 @ Y2 ) ) ).

% bezout_int
thf(fact_9789_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).

% gcd_le2_int
thf(fact_9790_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).

% gcd_le1_int
thf(fact_9791_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_9792_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A )
        & ( dvd_dvd_int @ D @ B )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A )
              & ( dvd_dvd_int @ E3 @ B ) )
           => ( dvd_dvd_int @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A @ B ) ) ) ).

% gcd_unique_int
thf(fact_9793_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_9794_VEBT__internal_Oheight_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_height @ X2 )
        = Y2 )
     => ( ( ? [A3: $o,B2: $o] :
              ( X2
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y2 != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.height.elims
thf(fact_9795_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M7: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K4: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K4 @ N ) @ M7 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9796_VEBT__internal_Oheight_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_height @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X2 )
       => ( ! [A3: $o,B2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y2 = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_9797_bin__last__integer__nbe,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I2: code_integer] :
          ( ( modulo364778990260209775nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_nbe
thf(fact_9798_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_9799_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9800_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9801_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_9802_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9803_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9804_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9805_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ N2 ) ) )
        = ( abs_abs_int @ N2 ) ) ) ).

% Max_divisors_self_int
thf(fact_9806_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_9807_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_9808_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9809_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9810_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).

% gcd_le2_nat
thf(fact_9811_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S8: set_int] :
        ? [K4: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S8 ) @ ( set_ord_lessThan_int @ K4 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9812_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S8: set_int] :
        ? [K4: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S8 ) @ ( set_ord_atMost_int @ K4 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9813_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9814_bitval__bin__last__integer,axiom,
    ! [I: code_integer] :
      ( ( zero_n356916108424825756nteger @ ( bits_b8758750999018896077nteger @ I ) )
      = ( bit_se3949692690581998587nteger @ I @ one_one_Code_integer ) ) ).

% bitval_bin_last_integer
thf(fact_9815_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] :
          ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9816_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9817_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N2 )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] :
                ( ( dvd_dvd_int @ D2 @ M )
                & ( dvd_dvd_int @ D2 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_9818_bin__last__integer__code,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I2: code_integer] :
          ( ( bit_se3949692690581998587nteger @ I2 @ one_one_Code_integer )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_code
thf(fact_9819_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_9820_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_9821_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_9822_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_9823_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_9824_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_9825_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M7: nat] : ( modulo_modulo_nat @ M7 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9826_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( gcd_gcd_nat @ M @ N2 )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M )
                & ( dvd_dvd_nat @ D2 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9827_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X: int] : ( plus_plus_int @ X @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9828_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X: code_integer] : ( plus_p5714425477246183910nteger @ X @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_9829_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_9830_bezw__aux,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X2 @ Y2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X2 @ Y2 ) ) @ ( semiri1314217659103216013at_int @ X2 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X2 @ Y2 ) ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) ) ).

% bezw_aux
thf(fact_9831_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y2: nat,X2: nat] :
      ( ( ( ord_less_nat @ C @ Y2 )
       => ( ( image_nat_nat
            @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ 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
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X2 @ Y2 )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9832_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_9833_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_9834_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_9835_bin__rest__integer__code,axiom,
    ( bits_b2549910563261871055nteger
    = ( ^ [I2: code_integer] : ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer_code
thf(fact_9836_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_9837_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_9838_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_9839_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_9840_wait__rule,axiom,
    ! [N2: nat] :
      ( hoare_8945653483474564448t_unit @ one_one_assn @ ( heap_Time_wait @ N2 )
      @ ^ [Uu3: product_unit] : one_one_assn ) ).

% wait_rule
thf(fact_9841_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M7: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M7 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M7 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9842_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_9843_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_9844_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9845_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_9846_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_9847_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9848_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_9849_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9850_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_9851_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9852_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_9853_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_9854_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_9855_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_9856_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N: nat,K4: int] : ( divide_divide_int @ K4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_int_def
thf(fact_9857_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N: nat,M7: nat] : ( divide_divide_nat @ M7 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9858_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_9859_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_9860_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_9861_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_9862_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_9863_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_9864_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M7: nat,N: nat] : ( bit_se1412395901928357646or_nat @ N @ ( bit_se547839408752420682it_nat @ M7 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9865_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_9866_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M7: nat,N: nat] : ( bit_se6528837805403552850or_nat @ N @ ( bit_se547839408752420682it_nat @ M7 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9867_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_9868_Bit__Operations_Oset__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K4: int] : ( bit_se1409905431419307370or_int @ K4 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% Bit_Operations.set_bit_int_def
thf(fact_9869_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_9870_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N: nat,K4: int] : ( bit_se6526347334894502574or_int @ K4 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9871_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_9872_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K4: int] : ( bit_se725231765392027082nd_int @ K4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9873_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N: nat,K4: int] : ( times_times_int @ K4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_int_def
thf(fact_9874_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_9875_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N: nat,M7: nat] : ( times_times_nat @ M7 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_nat_def
thf(fact_9876_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_9877_set__bit__integer__conv__masks,axiom,
    ( generi2397576812484419408nteger
    = ( ^ [X: code_integer,I2: nat,B4: $o] : ( if_Code_integer @ B4 @ ( bit_se1080825931792720795nteger @ X @ ( bit_se7788150548672797655nteger @ I2 @ one_one_Code_integer ) ) @ ( bit_se3949692690581998587nteger @ X @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ I2 @ one_one_Code_integer ) ) ) ) ) ) ).

% set_bit_integer_conv_masks
thf(fact_9878_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
thf(fact_9879_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_9880_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_9881_int__set__bit__conv__ops,axiom,
    ( generi8991105624351003935it_int
    = ( ^ [I2: int,N: nat,B4: $o] : ( if_int @ B4 @ ( bit_se1409905431419307370or_int @ I2 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) @ ( bit_se725231765392027082nd_int @ I2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ) ).

% int_set_bit_conv_ops
thf(fact_9882_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9883_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_9884_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_9885_int__lsb__numeral_I2_J,axiom,
    least_4859182151741483524sb_int @ one_one_int ).

% int_lsb_numeral(2)
thf(fact_9886_int__lsb__numeral_I6_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ).

% int_lsb_numeral(6)
thf(fact_9887_int__lsb__numeral_I3_J,axiom,
    least_4859182151741483524sb_int @ ( numeral_numeral_int @ one ) ).

% int_lsb_numeral(3)
thf(fact_9888_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) ).

% int_lsb_numeral(7)
thf(fact_9889_int__lsb__numeral_I4_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ one_one_int ) ).

% int_lsb_numeral(4)
thf(fact_9890_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_9891_int__lsb__numeral_I5_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ).

% int_lsb_numeral(5)
thf(fact_9892_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_9893_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_9894_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_9895_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_9896_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_9897_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_9898_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_9899_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9900_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_9901_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_9902_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9903_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_9904_upto_Osimps,axiom,
    ( upto
    = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9905_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_9906_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_9907_bin__last__conv__lsb,axiom,
    ( ( ^ [A4: int] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) )
    = least_4859182151741483524sb_int ) ).

% bin_last_conv_lsb
thf(fact_9908_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_9909_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_9910_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9911_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R3: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R3 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R3 ) ) ).

% concat_bit_assoc
thf(fact_9912_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K4: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ K4 ) @ ( bit_se545348938243370406it_int @ N @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_9913_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9914_assn__aci_I11_J,axiom,
    ! [X2: assn,Y2: assn,A: assn,B: assn] :
      ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X2 @ Y2 ) @ A )
     => ( ( times_times_assn @ A @ B )
        = ( times_times_assn @ B @ A ) ) ) ).

% assn_aci(11)
thf(fact_9915_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_9916_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_9917_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9918_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_9919_int__sdiv__simps_I1_J,axiom,
    ! [A: int] :
      ( ( signed6714573509424544716de_int @ A @ one_one_int )
      = A ) ).

% int_sdiv_simps(1)
thf(fact_9920_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_9921_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = A )
        = ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
thf(fact_9922_int__sdiv__simps_I3_J,axiom,
    ! [A: int] :
      ( ( signed6714573509424544716de_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% int_sdiv_simps(3)
thf(fact_9923_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_9924_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_9925_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_9926_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = ( uminus_uminus_int @ A ) )
        = ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_sdiv_negated_is_minus1
thf(fact_9927_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_9928_upt__eq__lel__conv,axiom,
    ! [L: nat,H2: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( ( upt @ L @ H2 )
        = ( append_nat @ Is1 @ ( cons_nat @ I @ Is2 ) ) )
      = ( ( Is1
          = ( upt @ L @ I ) )
        & ( Is2
          = ( upt @ ( suc @ I ) @ H2 ) )
        & ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ H2 ) ) ) ).

% upt_eq_lel_conv
thf(fact_9929_upt__rec,axiom,
    ( upt
    = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9930_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_9931_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_9932_map__Suc__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_9933_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_9934_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_9935_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_9936_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 ) )
        = ( ? [K4: nat] :
              ( ( ord_less_eq_nat @ I @ K4 )
              & ( ord_less_eq_nat @ K4 @ J )
              & ( ( upt @ I @ K4 )
                = Xs2 )
              & ( ( upt @ K4 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_9937_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I2: nat,J3: nat] : ( set_nat2 @ ( upt @ I2 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_9938_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_9939_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_9940_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_9941_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_9942_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_9943_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N: nat,M7: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M7 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_9944_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% atLeast_upt
thf(fact_9945_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( signed6714573509424544716de_int @ A @ B )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( signed6714573509424544716de_int @ A @ B ) )
        = ( sgn_sgn_int @ ( times_times_int @ A @ B ) ) ) ) ).

% sgn_sdiv_eq_sgn_mult
thf(fact_9946_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_9947_signed__divide__int__def,axiom,
    ( signed6714573509424544716de_int
    = ( ^ [K4: int,L2: int] : ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K4 ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K4 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% signed_divide_int_def
thf(fact_9948_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X5: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M7: nat] :
          ( ( ord_less_eq_nat @ M9 @ M7 )
         => ! [N: nat] :
              ( ( ord_less_eq_nat @ M9 @ N )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X5 @ M7 ) @ ( X5 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9949_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_9950_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_9951_fails__assert_H,axiom,
    ! [P: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( time_f8834461667527620124t_unit @ ( refine_Imp_assert @ P ) @ H2 )
      = ~ P ) ).

% fails_assert'
thf(fact_9952_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U4 )
     => ( ! [I3: nat] :
            ( ( ( ord_less_eq_nat @ U @ I3 )
              & ( ord_less_nat @ I3 @ U4 ) )
           => ~ ( P @ I3 ) )
       => ( ( filter_nat @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat @ P @ ( upt @ zero_zero_nat @ U4 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_9953_sort__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X: nat] : X
        @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sort_upt
thf(fact_9954_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X: int] : X
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_9955_tl__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( tl_nat @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ N2 ) ) ).

% tl_upt
thf(fact_9956_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_9957_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size(3)
thf(fact_9958_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size_gen(1)
thf(fact_9959_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ top_top_assn )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% entails_solve_init(1)
thf(fact_9960_FI__QUERY__def,axiom,
    ( fI_QUERY
    = ( ^ [P3: assn,Q4: assn,F7: assn] : ( entails @ P3 @ ( times_times_assn @ Q4 @ F7 ) ) ) ) ).

% FI_QUERY_def
thf(fact_9961_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F5: assn] :
      ( ( fI_QUERY @ P @ Q @ F5 )
     => ( entails @ P @ ( times_times_assn @ Q @ F5 ) ) ) ).

% frame_inference_init
thf(fact_9962_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ one_one_assn )
     => ( entails @ P @ Q ) ) ).

% entails_solve_init(2)
thf(fact_9963_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( member_int @ ( signed6292675348222524329lo_int @ A @ B ) @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( abs_abs_int @ B ) ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ B ) @ one_one_int ) ) ) ) ).

% smod_int_range
thf(fact_9964_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_9965_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_9966_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A @ B ) @ B ) ) ) ).

% smod_int_compares(1)
thf(fact_9967_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(2)
thf(fact_9968_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A @ B ) @ zero_zero_int ) ) ) ).

% smod_int_compares(4)
thf(fact_9969_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(6)
thf(fact_9970_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A @ B ) @ zero_zero_int ) ) ) ).

% smod_int_compares(7)
thf(fact_9971_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ B @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(8)
thf(fact_9972_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( signed6292675348222524329lo_int @ A @ B )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% smod_mod_positive
thf(fact_9973_signed__modulo__int__def,axiom,
    ( signed6292675348222524329lo_int
    = ( ^ [K4: int,L2: int] : ( minus_minus_int @ K4 @ ( times_times_int @ ( signed6714573509424544716de_int @ K4 @ L2 ) @ L2 ) ) ) ) ).

% signed_modulo_int_def
thf(fact_9974_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(3)
thf(fact_9975_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A @ B ) @ ( uminus_uminus_int @ B ) ) ) ) ).

% smod_int_compares(5)
thf(fact_9976_smod__int__alt__def,axiom,
    ( signed6292675348222524329lo_int
    = ( ^ [A4: int,B4: int] : ( times_times_int @ ( sgn_sgn_int @ A4 ) @ ( modulo_modulo_int @ ( abs_abs_int @ A4 ) @ ( abs_abs_int @ B4 ) ) ) ) ) ).

% smod_int_alt_def
thf(fact_9977_len__of__finite__1__def,axiom,
    ( type_l31302759751748491nite_1
    = ( ^ [X: itself_finite_1] : one_one_nat ) ) ).

% len_of_finite_1_def
thf(fact_9978_len__num1,axiom,
    ( type_l4264026598287037465l_num1
    = ( ^ [Uu4: itself_Numeral_num1] : one_one_nat ) ) ).

% len_num1
thf(fact_9979_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_9980_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_9981_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_9982_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9983_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9984_min__minus_H,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ K ) @ M )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus'
thf(fact_9985_min__minus,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ M @ ( minus_minus_nat @ M @ K ) )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus
thf(fact_9986_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ B @ ( suc @ A ) )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(2)
thf(fact_9987_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ ( suc @ A ) @ B )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(1)
thf(fact_9988_min__pm,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ A @ B ) @ ( minus_minus_nat @ A @ B ) )
      = A ) ).

% min_pm
thf(fact_9989_min__pm1,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ ( ord_min_nat @ A @ B ) )
      = A ) ).

% min_pm1
thf(fact_9990_rev__min__pm,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ B @ A ) @ ( minus_minus_nat @ A @ B ) )
      = A ) ).

% rev_min_pm
thf(fact_9991_rev__min__pm1,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ ( ord_min_nat @ B @ A ) )
      = A ) ).

% rev_min_pm1
thf(fact_9992_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_9993_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_9994_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_9995_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_9996_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_9997_concat__bit__assoc__sym,axiom,
    ! [M: nat,N2: nat,K: int,L: int,R3: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N2 @ K @ L ) @ R3 )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N2 ) @ L @ R3 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_9998_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N2 ) @ L ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_9999_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_10000_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_10001_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_10002_int__set__bits__K__False,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu3: nat] : $false )
    = zero_zero_int ) ).

% int_set_bits_K_False
thf(fact_10003_int__set__bits__K__True,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu3: nat] : $true )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_set_bits_K_True
thf(fact_10004_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_10005_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_10006_wf__set__bits__int__const,axiom,
    ! [B: $o] :
      ( bit_wf_set_bits_int
      @ ^ [Uu3: nat] : B ) ).

% wf_set_bits_int_const
thf(fact_10007_ones,axiom,
    ! [N2: nat,F: nat > $o] :
      ( ! [N7: nat] :
          ( ( ord_less_eq_nat @ N2 @ N7 )
         => ( F @ N7 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% ones
thf(fact_10008_wf__set__bits__int_Ocases,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ! [N4: nat] :
            ~ ! [N6: nat] :
                ( ( ord_less_eq_nat @ N4 @ N6 )
               => ~ ( F @ N6 ) )
       => ~ ! [N4: nat] :
              ~ ! [N6: nat] :
                  ( ( ord_less_eq_nat @ N4 @ N6 )
                 => ( F @ N6 ) ) ) ) ).

% wf_set_bits_int.cases
thf(fact_10009_wf__set__bits__int_Osimps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F4: nat > $o] :
          ( ? [N: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N @ N11 )
             => ~ ( F4 @ N11 ) )
          | ? [N: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N @ N11 )
             => ( F4 @ N11 ) ) ) ) ) ).

% wf_set_bits_int.simps
thf(fact_10010_zeros,axiom,
    ! [N2: nat,F: nat > $o] :
      ( ! [N7: nat] :
          ( ( ord_less_eq_nat @ N2 @ N7 )
         => ~ ( F @ N7 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% zeros
thf(fact_10011_wf__set__bits__int__simps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F4: nat > $o] :
        ? [N: nat] :
          ( ! [N11: nat] :
              ( ( ord_less_eq_nat @ N @ N11 )
             => ~ ( F4 @ N11 ) )
          | ! [N11: nat] :
              ( ( ord_less_eq_nat @ N @ N11 )
             => ( F4 @ N11 ) ) ) ) ) ).

% wf_set_bits_int_simps
thf(fact_10012_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_10013_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_10014_card_Ocomp__fun__commute__on,axiom,
    ( ( comp_nat_nat_nat @ suc @ suc )
    = ( comp_nat_nat_nat @ suc @ suc ) ) ).

% card.comp_fun_commute_on
thf(fact_10015_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).

% Code_Numeral.negative_def
thf(fact_10016_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_10017_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_10018_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_10019_msb__numeral_I1_J,axiom,
    ! [N2: num] :
      ~ ( most_s5051101344085556sb_int @ ( numeral_numeral_int @ N2 ) ) ).

% msb_numeral(1)
thf(fact_10020_msb__1,axiom,
    ~ ( most_s5051101344085556sb_int @ one_one_int ) ).

% msb_1
thf(fact_10021_msb__numeral_I2_J,axiom,
    ! [N2: num] : ( most_s5051101344085556sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% msb_numeral(2)
thf(fact_10022_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_10023_inj__Suc,axiom,
    ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).

% inj_Suc
thf(fact_10024_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_10025_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_10026_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_10027_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_10028_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_10029_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_10030_msb__uint32__code,axiom,
    ( most_s9063628576841037300uint32
    = ( ^ [X: uint32] : ( uint32_test_bit @ X @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% msb_uint32_code
thf(fact_10031_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_10032_nat__of__integer__numeral,axiom,
    ! [N2: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% nat_of_integer_numeral
thf(fact_10033_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_10034_nat__of__integer__1,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_1
thf(fact_10035_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_10036_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_10037_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_10038_integer__set__bit__code,axiom,
    ( bits_integer_set_bit
    = ( ^ [X: code_integer,N: code_integer,B4: $o] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger ) @ ( undefi1878487536576149250nteger @ X @ N @ B4 ) @ ( if_Code_integer @ B4 @ ( 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_10039_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_10040_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_10041_uint32__set__bit__def,axiom,
    ( uint32_set_bit
    = ( ^ [X: uint32,N: code_integer,B4: $o] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
          @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ X @ N @ B4 )
          @ ( generi1993664874377053279uint32 @ X @ ( code_nat_of_integer @ N ) @ B4 ) ) ) ) ).

% uint32_set_bit_def
thf(fact_10042_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K4: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K4 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_10043_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
      = ( divide6298287555418463151nteger @ K @ L ) ) ).

% fst_divmod_integer
thf(fact_10044_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L ) )
      = ( modulo364778990260209775nteger @ K @ L ) ) ).

% snd_divmod_integer
thf(fact_10045_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K4 @ L2 ) @ ( modulo364778990260209775nteger @ K4 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_10046_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K4: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K4 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_10047_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K4: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K4 ) ) )
          @ ( if_int @ ( K4 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_10048_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_10049_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_10050_plus__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X2 @ Xa ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% plus_integer.rep_eq
thf(fact_10051_minus__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X2 @ Xa ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% minus_integer.rep_eq
thf(fact_10052_times__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X2 @ Xa ) )
      = ( times_times_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% times_integer.rep_eq
thf(fact_10053_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_10054_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K4: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_10055_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_10056_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_10057_dup_Orep__eq,axiom,
    ! [X2: code_integer] :
      ( ( code_int_of_integer @ ( code_dup @ X2 ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ X2 ) ) ) ).

% dup.rep_eq
thf(fact_10058_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_10059_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_10060_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_10061_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_10062_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_10063_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_10064_uint32_Oword__of__numeral,axiom,
    ! [N2: num] :
      ( ( rep_uint322 @ ( numera9087168376688890119uint32 @ N2 ) )
      = ( numera7442385471795722001l_num1 @ N2 ) ) ).

% uint32.word_of_numeral
thf(fact_10065_size__uint32_Orep__eq,axiom,
    ( size_size_uint32
    = ( ^ [X: uint32] : ( size_s8261804613246490634l_num1 @ ( rep_uint322 @ X ) ) ) ) ).

% size_uint32.rep_eq
thf(fact_10066_uint32_Osize__eq__word__of,axiom,
    ( size_size_uint32
    = ( ^ [P5: uint32] : ( size_s8261804613246490634l_num1 @ ( rep_uint322 @ P5 ) ) ) ) ).

% uint32.size_eq_word_of
thf(fact_10067_uint32_Oeven__iff__word__of,axiom,
    ! [P4: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ P4 )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( rep_uint322 @ P4 ) ) ) ).

% uint32.even_iff_word_of
thf(fact_10068_times__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( times_times_uint32 @ X2 @ Xa ) )
      = ( times_7065122842183080059l_num1 @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% times_uint32.rep_eq
thf(fact_10069_uint32_Oword__of__mult,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( times_times_uint32 @ P4 @ Q2 ) )
      = ( times_7065122842183080059l_num1 @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_mult
thf(fact_10070_divide__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( divide_divide_uint32 @ X2 @ Xa ) )
      = ( divide1791077408188789448l_num1 @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% divide_uint32.rep_eq
thf(fact_10071_uint32_Oword__of__div,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( divide_divide_uint32 @ P4 @ Q2 ) )
      = ( divide1791077408188789448l_num1 @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_div
thf(fact_10072_one__uint32_Orep__eq,axiom,
    ( ( rep_uint322 @ one_one_uint32 )
    = one_on7727431528512463931l_num1 ) ).

% one_uint32.rep_eq
thf(fact_10073_uint32_Oword__of__diff,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( minus_minus_uint32 @ P4 @ Q2 ) )
      = ( minus_4019991460397169231l_num1 @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_diff
thf(fact_10074_minus__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( minus_minus_uint32 @ X2 @ Xa ) )
      = ( minus_4019991460397169231l_num1 @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% minus_uint32.rep_eq
thf(fact_10075_plus__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( plus_plus_uint32 @ X2 @ Xa ) )
      = ( plus_p361126936061061375l_num1 @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% plus_uint32.rep_eq
thf(fact_10076_uint32_Oword__of__add,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( plus_plus_uint32 @ P4 @ Q2 ) )
      = ( plus_p361126936061061375l_num1 @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_add
thf(fact_10077_uint32_Oword__of__power,axiom,
    ! [P4: uint32,N2: nat] :
      ( ( rep_uint322 @ ( power_power_uint32 @ P4 @ N2 ) )
      = ( power_2184487114949457152l_num1 @ ( rep_uint322 @ P4 ) @ N2 ) ) ).

% uint32.word_of_power
thf(fact_10078_less__eq__uint32_Orep__eq,axiom,
    ( ord_less_eq_uint32
    = ( ^ [X: uint32,Xa4: uint32] : ( ord_le3335648743751981014l_num1 @ ( rep_uint322 @ X ) @ ( rep_uint322 @ Xa4 ) ) ) ) ).

% less_eq_uint32.rep_eq
thf(fact_10079_uint32_Oless__eq__iff__word__of,axiom,
    ( ord_less_eq_uint32
    = ( ^ [P5: uint32,Q5: uint32] : ( ord_le3335648743751981014l_num1 @ ( rep_uint322 @ P5 ) @ ( rep_uint322 @ Q5 ) ) ) ) ).

% uint32.less_eq_iff_word_of
thf(fact_10080_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_10081_uint32__set__bit__code,axiom,
    ! [N2: code_integer,W: uint32,B: $o] :
      ( ( ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
          | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N2 @ B ) )
          = ( rep_uint322 @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ W @ N2 @ B ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N2 @ B ) )
          = ( generi5268133209446125161l_num1 @ ( rep_uint322 @ W ) @ ( code_nat_of_integer @ N2 ) @ B ) ) ) ) ).

% uint32_set_bit_code
thf(fact_10082_set__bit__uint32__code,axiom,
    ( generi1993664874377053279uint32
    = ( ^ [X: uint32,N: nat,B4: $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 ) @ B4 ) @ X ) ) ) ).

% set_bit_uint32_code
thf(fact_10083_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_10084_one__uint32_Orsp,axiom,
    one_on7727431528512463931l_num1 = one_on7727431528512463931l_num1 ).

% one_uint32.rsp
thf(fact_10085_integer__of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% integer_of_nat_numeral
thf(fact_10086_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_10087_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_10088_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_10089_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_10090_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_10091_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_10092_uint32_Oset__bits__code,axiom,
    ( bit_bi705532357378895591uint32
    = ( ^ [P3: nat > $o] : ( set_bits_aux_uint32 @ P3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ zero_zero_uint32 ) ) ) ).

% uint32.set_bits_code
thf(fact_10093_div0__uint32__def,axiom,
    ( div0_uint32
    = ( ^ [X: uint32] : ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X @ zero_zero_uint32 ) ) ) ).

% div0_uint32_def
thf(fact_10094_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_10095_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_10096_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_10097_uint32__div__def,axiom,
    ( uint32_div
    = ( ^ [X: uint32,Y: uint32] : ( produc9004433772639906525uint32 @ ( uint32_divmod @ X @ Y ) ) ) ) ).

% uint32_div_def
thf(fact_10098_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_10099_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_10100_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_10101_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_10102_less__eq__uint32_Oabs__eq,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( ord_le3335648743751981014l_num1 @ Xa @ X2 ) ) ).

% less_eq_uint32.abs_eq
thf(fact_10103_plus__uint32_Oabs__eq,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( plus_plus_uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( plus_p361126936061061375l_num1 @ Xa @ X2 ) ) ) ).

% plus_uint32.abs_eq
thf(fact_10104_minus__uint32_Oabs__eq,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( minus_minus_uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( minus_4019991460397169231l_num1 @ Xa @ X2 ) ) ) ).

% minus_uint32.abs_eq
thf(fact_10105_one__uint32__def,axiom,
    ( one_one_uint32
    = ( abs_uint32 @ one_on7727431528512463931l_num1 ) ) ).

% one_uint32_def
thf(fact_10106_divide__uint32_Oabs__eq,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( divide_divide_uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( divide1791077408188789448l_num1 @ Xa @ X2 ) ) ) ).

% divide_uint32.abs_eq
thf(fact_10107_times__uint32_Oabs__eq,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( times_times_uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( times_7065122842183080059l_num1 @ Xa @ X2 ) ) ) ).

% times_uint32.abs_eq
thf(fact_10108_uint32_Oof__word__numeral,axiom,
    ! [N2: num] :
      ( ( abs_uint32 @ ( numera7442385471795722001l_num1 @ N2 ) )
      = ( numera9087168376688890119uint32 @ N2 ) ) ).

% uint32.of_word_numeral
thf(fact_10109_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_10110_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_10111_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_10112_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K4 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K4 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S4 ) ) )
                  @ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_10113_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L ) )
      = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).

% fst_divmod_abs
thf(fact_10114_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L ) )
      = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).

% snd_divmod_abs
thf(fact_10115_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_10116_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_10117_Rep__uint32_H__code,axiom,
    ( rep_uint32
    = ( ^ [X: uint32] : ( bit_bi5746210779246519537l_num1 @ ( bit_se5367290876889521763uint32 @ X ) ) ) ) ).

% Rep_uint32'_code
thf(fact_10118_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K4: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_10119_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ ( code_divmod_abs @ K4 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S4 ) ) )
                @ ( code_divmod_abs @ K4 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K4 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S4 ) ) )
                    @ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_10120_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ R @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_10121_set__bits__int__def,axiom,
    ( bit_bi6516823479961619367ts_int
    = ( ^ [F4: nat > $o] :
          ( if_int
          @ ? [N: nat] :
            ! [M7: nat] :
              ( ( ord_less_eq_nat @ N @ M7 )
             => ( ( F4 @ M7 )
                = ( F4 @ N ) ) )
          @ ( bit_ri631733984087533419it_int
            @ ( ord_Least_nat
              @ ^ [N: nat] :
                ! [M7: nat] :
                  ( ( ord_less_eq_nat @ N @ M7 )
                 => ( ( F4 @ M7 )
                    = ( 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] :
                        ! [M7: nat] :
                          ( ( ord_less_eq_nat @ N @ M7 )
                         => ( ( F4 @ M7 )
                            = ( F4 @ N ) ) ) ) ) ) ) ) )
          @ zero_zero_int ) ) ) ).

% set_bits_int_def
thf(fact_10122_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_10123_Least__Suc,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M7: nat] : ( P @ ( suc @ M7 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_10124_Least__Suc2,axiom,
    ! [P: nat > $o,N2: nat,Q: nat > $o,M: nat] :
      ( ( P @ N2 )
     => ( ( Q @ M )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K3: nat] :
                ( ( P @ ( suc @ K3 ) )
                = ( Q @ K3 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_10125_set__bits__int__unfold_H,axiom,
    ( bit_bi6516823479961619367ts_int
    = ( ^ [F4: nat > $o] :
          ( if_int
          @ ? [N: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N @ N11 )
             => ~ ( F4 @ N11 ) )
          @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
            @ ( map_nat_o @ F4
              @ ( upt @ zero_zero_nat
                @ ( ord_Least_nat
                  @ ^ [N: nat] :
                    ! [N11: nat] :
                      ( ( ord_less_eq_nat @ N @ N11 )
                     => ~ ( F4 @ N11 ) ) ) ) ) )
          @ ( if_int
            @ ? [N: nat] :
              ! [N11: nat] :
                ( ( ord_less_eq_nat @ N @ N11 )
               => ( F4 @ N11 ) )
            @ ( bit_ri631733984087533419it_int
              @ ( ord_Least_nat
                @ ^ [N: nat] :
                  ! [N11: nat] :
                    ( ( ord_less_eq_nat @ N @ N11 )
                   => ( F4 @ N11 ) ) )
              @ ( 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] :
                          ! [N11: nat] :
                            ( ( ord_less_eq_nat @ N @ N11 )
                           => ( F4 @ N11 ) ) ) ) )
                  @ ( cons_o @ $true @ nil_o ) ) ) )
            @ zero_zero_int ) ) ) ) ).

% set_bits_int_unfold'
thf(fact_10126_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K4 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_10127_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( Deg2 = Xa )
                & ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                   => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I2: nat] :
                            ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                                 => ( ( ord_less_nat @ Mi3 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_10128_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_10129_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_10130_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_10131_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_10132_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima3: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg6: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima3 @ Deg @ TreeList @ Summary ) @ Deg6 )
      = ( ( Deg = Deg6 )
        & ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
           => ( 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 @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X5 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima3 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_10133_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y2
            = ( Xa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
                = ( ~ ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_10134_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa )
                  & ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                                   => ( ( ord_less_nat @ Mi3 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_10135_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y2
                  = ( Xa = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_10136_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
               => ( Xa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
                 => ~ ( ( Deg2 = Xa )
                      & ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_10137_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_10138_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 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
               => ( Xa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
                 => ( ( Deg2 = Xa )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I2: nat] :
                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                                     => ( ( ord_less_nat @ Mi3 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_10139_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_10140_FI__def,axiom,
    ( fi
    = ( ^ [M7: list_P8527749157015355191n_assn,P5: assn,Q5: assn,Up: assn,Uq: assn,F4: assn] :
          ( ! [X: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ M7 ) )
             => ( produc7274209992780475162assn_o @ entails @ X ) )
         => ( entails @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M7 ) @ 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 @ M7 ) @ one_one_assn ) @ Q5 ) @ Uq ) @ F4 ) ) ) ) ) ).

% FI_def
thf(fact_10141_entails__solve__finalize_I2_J,axiom,
    ! [M8: list_P8527749157015355191n_assn] : ( fI_RESULT @ M8 @ one_one_assn @ one_one_assn @ one_one_assn ) ).

% entails_solve_finalize(2)
thf(fact_10142_frame__inference__finalize,axiom,
    ! [M8: list_P8527749157015355191n_assn,F5: assn] : ( fI_RESULT @ M8 @ F5 @ one_one_assn @ F5 ) ).

% frame_inference_finalize
thf(fact_10143_FI__p__nomatch,axiom,
    ! [M: list_P8527749157015355191n_assn,Ps2: assn,Qs: assn,Q2: assn,P4: assn,Up2: assn,Uq2: assn,F: assn] :
      ( ( fi @ M @ Ps2 @ ( times_times_assn @ Qs @ Q2 ) @ ( times_times_assn @ P4 @ Up2 ) @ Uq2 @ F )
     => ( fi @ M @ ( times_times_assn @ Ps2 @ P4 ) @ ( times_times_assn @ Qs @ Q2 ) @ Up2 @ Uq2 @ F ) ) ).

% FI_p_nomatch
thf(fact_10144_FI__finalize,axiom,
    ! [M: list_P8527749157015355191n_assn,P4: assn,Up2: assn,Q2: assn,Uq2: assn,F: assn] :
      ( ( fI_RESULT @ M @ ( times_times_assn @ P4 @ Up2 ) @ ( times_times_assn @ Q2 @ Uq2 ) @ F )
     => ( fi @ M @ P4 @ Q2 @ Up2 @ Uq2 @ F ) ) ).

% FI_finalize
thf(fact_10145_entails__solve__finalize_I1_J,axiom,
    ! [M8: list_P8527749157015355191n_assn,P: assn] : ( fI_RESULT @ M8 @ P @ one_one_assn @ top_top_assn ) ).

% entails_solve_finalize(1)
thf(fact_10146_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,C52: $o,C62: $o,C7: $o] :
      ( ( ord_less_eq_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C7 ) )
      = ( ord_less_eq_nat
        @ ( foldr_o_nat
          @ ^ [B4: $o,K4: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K4 @ ( 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
          @ ^ [B4: $o,K4: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C62 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_eq_char_simp
thf(fact_10147_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,C52: $o,C62: $o,C7: $o] :
      ( ( ord_less_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C7 ) )
      = ( ord_less_nat
        @ ( foldr_o_nat
          @ ^ [B4: $o,K4: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K4 @ ( 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
          @ ^ [B4: $o,K4: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C62 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_char_simp
thf(fact_10148_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_10149_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_10150_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_10151_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_10152_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_10153_char_Osize_I2_J,axiom,
    ! [X1: $o,X23: $o,X33: $o,X43: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X23 @ X33 @ X43 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_10154_char__of__integer__code,axiom,
    ( char_of_integer
    = ( ^ [K4: code_integer] :
          ( produc4188289175737317920o_char
          @ ^ [Q0: code_integer,B02: $o] :
              ( produc4188289175737317920o_char
              @ ^ [Q1: code_integer,B12: $o] :
                  ( produc4188289175737317920o_char
                  @ ^ [Q22: code_integer,B23: $o] :
                      ( produc4188289175737317920o_char
                      @ ^ [Q32: code_integer,B33: $o] :
                          ( produc4188289175737317920o_char
                          @ ^ [Q42: code_integer,B43: $o] :
                              ( produc4188289175737317920o_char
                              @ ^ [Q52: code_integer,B53: $o] :
                                  ( produc4188289175737317920o_char
                                  @ ^ [Q62: code_integer,B63: $o] :
                                      ( produc4188289175737317920o_char
                                      @ ^ [Uu3: code_integer] : ( char2 @ B02 @ B12 @ B23 @ B33 @ B43 @ B53 @ B63 )
                                      @ ( code_bit_cut_integer @ Q62 ) )
                                  @ ( code_bit_cut_integer @ Q52 ) )
                              @ ( code_bit_cut_integer @ Q42 ) )
                          @ ( code_bit_cut_integer @ Q32 ) )
                      @ ( code_bit_cut_integer @ Q22 ) )
                  @ ( code_bit_cut_integer @ Q1 ) )
              @ ( code_bit_cut_integer @ Q0 ) )
          @ ( code_bit_cut_integer @ K4 ) ) ) ) ).

% char_of_integer_code
thf(fact_10155_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_10156_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_10157_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_10158_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_10159_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X2: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( minus_minus_real @ X2 @ H6 ) @ S2 )
                 => ( ( ord_less_real @ H6 @ D3 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_10160_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X2: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( minus_minus_real @ X2 @ H6 ) @ S2 )
                 => ( ( ord_less_real @ H6 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_10161_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X2: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( plus_plus_real @ X2 @ H6 ) @ S2 )
                 => ( ( ord_less_real @ H6 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_10162_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X2: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( plus_plus_real @ X2 @ H6 ) @ S2 )
                 => ( ( ord_less_real @ H6 @ D3 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_10163_DERIV__pow,axiom,
    ! [N2: nat,X2: real,S3: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ).

% DERIV_pow
thf(fact_10164_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_10165_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_10166_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_10167_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_10168_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_10169_DERIV__const__average,axiom,
    ! [A: real,B: real,V: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_10170_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_10171_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
          = K ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_10172_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_10173_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D3 )
                 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_10174_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_10175_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_10176_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D3 )
                 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_10177_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( 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 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const

% Helper facts (54)
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 ) )
      = ( ? [X5: real] : ( P @ X5 ) ) ) ).

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__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__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_T,axiom,
    ! [X2: heap_T5738788834812785303t_unit,Y2: heap_T5738788834812785303t_unit] :
      ( ( if_Hea8138950348631371857t_unit @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_T,axiom,
    ! [X2: heap_T5738788834812785303t_unit,Y2: heap_T5738788834812785303t_unit] :
      ( ( if_Hea8138950348631371857t_unit @ $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 (3)
thf(conj_0,hypothesis,
    ( tia
    = ( vEBT_Nodei @ x11 @ x12 @ x13 @ x14 ) ) ).

thf(conj_1,hypothesis,
    ( ( x11
      = ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) )
    & ( x12
      = ( suc @ ( suc @ va ) ) ) ) ).

thf(conj_2,conjecture,
    ( hoare_7629718768684598413on_nat @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ summary @ x14 ) @ ( snga_assn_VEBT_VEBTi @ x13 @ tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ treeList @ tree_is ) )
    @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ $true )
      @ ^ [Uu3: product_unit] :
          ( heap_T5999496708990702694on_nat
          @ ( refine_Imp_assert
            @ ( ( ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) )
                = x11 )
              & ( ( suc @ ( suc @ va ) )
                = x12 ) ) )
          @ ^ [Uv3: product_unit] :
              ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
              @ ^ [Mima2: product_prod_nat_nat] :
                  ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ x12 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima2 ) @ xa ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima2 ) ) )
                    @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ xa @ ( divide_divide_nat @ x12 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                      @ ^ [L2: nat] :
                          ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ xa @ ( divide_divide_nat @ x12 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                          @ ^ [H: nat] :
                              ( heap_T5999496708990702694on_nat
                              @ ( refine_Imp_assert
                                @ ( L2
                                  = ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ x12 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              @ ^ [Uw3: product_unit] :
                                  ( heap_T5999496708990702694on_nat
                                  @ ( refine_Imp_assert
                                    @ ( H
                                      = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ x12 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ^ [Ux3: product_unit] :
                                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ treeList ) ) )
                                      @ ^ [Uy3: product_unit] :
                                          ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ x13 @ 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 ) ) ) )
                                                  @ ^ [Uz3: product_unit] :
                                                      ( if_Hea5867803462524415986on_nat
                                                      @ ( ( Minlow != none_nat )
                                                        & ( vEBT_VEBT_greater @ ( some_nat @ L2 ) @ Minlow ) )
                                                      @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_predi @ ( nth_VEBT_VEBT @ treeList @ H ) @ Aktnode @ L2 )
                                                        @ ^ [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 @ x12 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                      @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_predi @ summary @ x14 @ H )
                                                        @ ^ [Predsum: option_nat] :
                                                            ( heap_T5999496708990702694on_nat
                                                            @ ( refine_Imp_assert
                                                              @ ( ( Predsum = none_nat )
                                                                = ( ( vEBT_vebt_pred @ summary @ H )
                                                                  = none_nat ) ) )
                                                            @ ^ [Va4: product_unit] :
                                                                ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima2 ) @ xa ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima2 ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ x13 @ ( 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 @ x12 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ x11 ) ) )
    @ ^ [R: option_nat] :
        ( ex_ass463751140784270563_VEBTi
        @ ^ [Tree_is: list_VEBT_VEBTi] :
            ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ summary @ x14 ) @ ( snga_assn_VEBT_VEBTi @ x13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ treeList @ Tree_is ) )
            @ ( pure_assn
              @ ( ( x11
                  = ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) )
                & ( x12
                  = ( suc @ ( suc @ va ) ) )
                & ( ( ord_less_nat @ ma @ xa )
                 => ( R
                    = ( some_nat @ ma ) ) )
                & ( ~ ( ord_less_nat @ ma @ xa )
                 => ( R
                    = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
                      @ ( if_option_nat
                        @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           != none_nat )
                          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( if_option_nat
                          @ ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                            = none_nat )
                          @ ( if_option_nat @ ( ord_less_nat @ mi @ xa ) @ ( some_nat @ mi ) @ none_nat )
                          @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                      @ none_nat ) ) ) ) ) ) ) ) ).

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