%------------------------------------------------------------------------------
% File     : ITP292^3 : TPTP v9.0.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Intf_Imperative 00161_004904
% 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    : 0096_VEBT_Intf_Imperative_00161_004904 [Des22]

% Status   : Theorem
% Rating   : 0.88 v9.0.0, 1.00 v8.1.0
% Syntax   : Number of formulae    : 10921 (3598 unt;1771 typ;   0 def)
%            Number of atoms       : 31412 (11678 equ;   2 cnn)
%            Maximal formula atoms :   71 (   3 avg)
%            Number of connectives : 126270 (3401   ~; 433   |;2629   &;105382   @)
%                                         (   0 <=>;14425  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   7 avg)
%            Number of types       :  211 ( 210 usr)
%            Number of type conns  : 6094 (6094   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1566 (1561 usr;  83 con; 0-8 aty)
%            Number of variables   : 27945 (2245   ^;24428   !;1272   ?;27945   :)
% 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 23:43:13.532
%------------------------------------------------------------------------------
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    pure_assn: $o > assn ).

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

thf(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re3376528473927230327_nat_o: ( int > int > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( int > nat > $o ) > ( int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re3715656647883201625at_nat: ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > ( int > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
    bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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_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_001_062_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_M_Eo_J_001_062_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_M_Eo_J,type,
    bNF_re7364608769721783435num1_o: ( word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $o ) > ( ( word_N3645301735248828278l_num1 > $o ) > ( word_N3645301735248828278l_num1 > $o ) > $o ) > ( word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $o ) > ( word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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_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_001_Eo_001_Eo,type,
    bNF_re5013357767504289739m1_o_o: ( word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $o ) > ( $o > $o > $o ) > ( word_N3645301735248828278l_num1 > $o ) > ( word_N3645301735248828278l_num1 > $o ) > $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_Owf__set__bits__int,type,
    bit_wf_set_bits_int: ( nat > $o ) > $o ).

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

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_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_Omask_001t__Code____Numeral__Ointeger,type,
    bit_se2119862282449309892nteger: nat > code_integer ).

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_Otake__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1745604003318907178nteger: nat > code_integer > code_integer ).

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_Oxor_001t__Code____Numeral__Ointeger,type,
    bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Code__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_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_Ointeger__of__num,type,
    code_integer_of_num: num > 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_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
    complete_Sup_Sup_int: set_int > int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
    times_times_rat: rat > rat > rat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Uint32__Ouint32,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
    uminus_uminus_real_o: ( real > $o ) > real > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Assertions__Oassn,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: rat > rat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
    uminus_uminus_real: real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    uminus8566677241136511917omplex: set_complex > set_complex ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Uint32__Ouint32,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Uint32__Ouint32,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__String__Oliteral_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__String__Oliteral_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__String__Oliteral_001t__Nat__Onat,type,
    groups8652099787943017962al_nat: ( literal > nat ) > set_literal > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__String__Oliteral_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__String__Oliteral_001t__Uint32__Ouint32,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Uint32__Ouint32,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
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thf(sy_c_HOL_OThe_001t__Int__Oint,type,
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thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
    the_real: ( real > $o ) > real ).

thf(sy_c_HOL_Oundefined_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
    undefi8952517107220742160uint32: ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ).

thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Code____Numeral__Ointeger_J_J_J,type,
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thf(sy_c_Heap_Oarray_Osize__array_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,type,
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thf(sy_c_Heap__Time__Monad_Oreturn_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Nat__Onat,type,
    hoare_3067605981109127869le_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_If_001t__Assertions__Oassn,type,
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thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_If_001t__Complex__Ocomplex,type,
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thf(sy_c_If_001t__Int__Oint,type,
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thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    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,
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thf(sy_c_If_001t__Num__Onum,type,
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thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Set__Oset_I_Eo_J_J,type,
    if_option_set_o: $o > option_set_o > option_set_o > option_set_o ).

thf(sy_c_If_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J,type,
    if_option_set_int: $o > option_set_int > option_set_int > option_set_int ).

thf(sy_c_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    if_option_set_nat: $o > option_set_nat > option_set_nat > option_set_nat ).

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____Definitions__OVEBT,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
    infini8530281810654367211te_nat: set_nat > nat > nat ).

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

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

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

thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
    ring_1_Ints_int: set_int ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
    ring_1_Ints_rat: set_rat ).

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_OInts_001t__Uint32__Ouint32,type,
    ring_1_Ints_uint32: set_uint32 ).

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__Rat__Orat,type,
    ring_1_of_int_rat: int > rat ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Assertions__Oassn,type,
    inf_inf_assn: assn > assn > assn ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Assertions__Oassn,type,
    sup_sup_assn: assn > assn > assn ).

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

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

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001_Eo,type,
    lattic1921953407002678535_Max_o: set_o > $o ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Code____Numeral__Ointeger,type,
    lattic4901227151466704046nteger: set_Code_integer > code_integer ).

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_Lattices__Big_Olinorder__class_OMax_001t__Num__Onum,type,
    lattic4823215512031491691ax_num: set_num > num ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Rat__Orat,type,
    lattic7630753665789217321ax_rat: set_rat > rat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Real__Oreal,type,
    lattic4275903605611617917x_real: set_real > real ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001_Eo,type,
    lattic1973801136483472281_Min_o: set_o > $o ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Code____Numeral__Ointeger,type,
    lattic1063845414844153500nteger: set_Code_integer > code_integer ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
    lattic8718645017227715691in_int: set_int > int ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
    lattic8721135487736765967in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Num__Onum,type,
    lattic5278467273892544601in_num: set_num > num ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Rat__Orat,type,
    lattic8086005427650270231in_rat: set_rat > rat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Real__Oreal,type,
    lattic3629708407755379051n_real: set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Rat__Orat,type,
    lattic2140725968369957399_o_rat: ( $o > rat ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Real__Oreal,type,
    lattic8697145971487455083o_real: ( $o > real ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    lattic5439806495466278992er_rat: ( code_integer > rat ) > set_Code_integer > code_integer ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Code____Numeral__Ointeger_001t__Real__Oreal,type,
    lattic2659822949269061924r_real: ( code_integer > real ) > set_Code_integer > code_integer ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
    lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
    lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
    lattic2675449441010098035t_real: ( int > real ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
    lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
    lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
    lattic1613168225601753569al_num: ( real > num ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
    lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__VEBT____Definitions__OVEBT_001t__Num__Onum,type,
    lattic3331990488459210229BT_num: ( vEBT_VEBT > num ) > set_VEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
    lattic6139528642216935859BT_rat: ( vEBT_VEBT > rat ) > set_VEBT_VEBT > vEBT_VEBT ).

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

thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

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_Obutlast_001_Eo,type,
    butlast_o: list_o > list_o ).

thf(sy_c_List_Obutlast_001t__Int__Oint,type,
    butlast_int: list_int > list_int ).

thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
    butlast_nat: list_nat > list_nat ).

thf(sy_c_List_Obutlast_001t__Real__Oreal,type,
    butlast_real: list_real > list_real ).

thf(sy_c_List_Obutlast_001t__VEBT____BuildupMemImp__OVEBTi,type,
    butlast_VEBT_VEBTi: list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Obutlast_001t__VEBT____Definitions__OVEBT,type,
    butlast_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Oconcat_001_Eo,type,
    concat_o: list_list_o > list_o ).

thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
    concat_nat: list_list_nat > list_nat ).

thf(sy_c_List_Oconcat_001t__Real__Oreal,type,
    concat_real: list_list_real > list_real ).

thf(sy_c_List_Oconcat_001t__VEBT____Definitions__OVEBT,type,
    concat_VEBT_VEBT: list_list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Ocount__list_001_Eo,type,
    count_list_o: list_o > $o > nat ).

thf(sy_c_List_Ocount__list_001t__Code____Numeral__Ointeger,type,
    count_3970941599679287265nteger: list_Code_integer > code_integer > nat ).

thf(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
    count_list_complex: list_complex > complex > nat ).

thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
    count_list_int: list_int > int > nat ).

thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
    count_list_nat: list_nat > nat > nat ).

thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
    count_list_real: list_real > real > nat ).

thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
    count_list_set_nat: list_set_nat > set_nat > nat ).

thf(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
    count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).

thf(sy_c_List_Odistinct_001_Eo,type,
    distinct_o: list_o > $o ).

thf(sy_c_List_Odistinct_001t__Code____Numeral__Ointeger,type,
    distin1543349897113766820nteger: list_Code_integer > $o ).

thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
    distinct_complex: list_complex > $o ).

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

thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

thf(sy_c_List_Odistinct_001t__Num__Onum,type,
    distinct_num: list_num > $o ).

thf(sy_c_List_Odistinct_001t__Rat__Orat,type,
    distinct_rat: list_rat > $o ).

thf(sy_c_List_Odistinct_001t__Real__Oreal,type,
    distinct_real: list_real > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
    distinct_set_nat: list_set_nat > $o ).

thf(sy_c_List_Odistinct_001t__VEBT____BuildupMemImp__OVEBTi,type,
    distinct_VEBT_VEBTi: list_VEBT_VEBTi > $o ).

thf(sy_c_List_Odistinct_001t__VEBT____Definitions__OVEBT,type,
    distinct_VEBT_VEBT: list_VEBT_VEBT > $o ).

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_nat2: ( nat > $o ) > list_nat > list_nat ).

thf(sy_c_List_Ofoldl_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    foldl_assn_assn: ( assn > assn > assn ) > assn > list_assn > assn ).

thf(sy_c_List_Ofoldl_001t__Int__Oint_001t__Int__Oint,type,
    foldl_int_int: ( int > int > int ) > int > list_int > int ).

thf(sy_c_List_Ofoldl_001t__Nat__Onat_001_Eo,type,
    foldl_nat_o: ( nat > $o > nat ) > nat > list_o > nat ).

thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Nat__Onat,type,
    foldl_nat_nat: ( nat > nat > nat ) > nat > list_nat > nat ).

thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Real__Oreal,type,
    foldl_nat_real: ( nat > real > nat ) > nat > list_real > nat ).

thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    foldl_nat_VEBT_VEBT: ( nat > vEBT_VEBT > nat ) > nat > list_VEBT_VEBT > nat ).

thf(sy_c_List_Ofoldl_001t__Rat__Orat_001t__Rat__Orat,type,
    foldl_rat_rat: ( rat > rat > rat ) > rat > list_rat > rat ).

thf(sy_c_List_Ofoldl_001t__Real__Oreal_001t__Real__Oreal,type,
    foldl_real_real: ( real > real > real ) > real > list_real > real ).

thf(sy_c_List_Ofoldl_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    foldl_uint32_uint32: ( uint32 > uint32 > uint32 ) > uint32 > list_uint32 > uint32 ).

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__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_Ofoldr_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    foldr_VEBT_VEBT_nat: ( vEBT_VEBT > nat > nat ) > list_VEBT_VEBT > nat > nat ).

thf(sy_c_List_Olast_001_Eo,type,
    last_o: list_o > $o ).

thf(sy_c_List_Olast_001t__Int__Oint,type,
    last_int: list_int > int ).

thf(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

thf(sy_c_List_Olast_001t__Real__Oreal,type,
    last_real: list_real > real ).

thf(sy_c_List_Olast_001t__VEBT____BuildupMemImp__OVEBTi,type,
    last_VEBT_VEBTi: list_VEBT_VEBTi > vEBT_VEBTi ).

thf(sy_c_List_Olast_001t__VEBT____Definitions__OVEBT,type,
    last_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT ).

thf(sy_c_List_Olenlex_001_Eo,type,
    lenlex_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

thf(sy_c_List_Olenlex_001t__Int__Oint,type,
    lenlex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
    lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olenlex_001t__Real__Oreal,type,
    lenlex_real: set_Pr6218003697084177305l_real > set_Pr611352295856513593t_real ).

thf(sy_c_List_Olenlex_001t__Uint32__Ouint32,type,
    lenlex_uint32: set_Pr1773385645901665561uint32 > set_Pr2258164808687509945uint32 ).

thf(sy_c_List_Olenlex_001t__VEBT____Definitions__OVEBT,type,
    lenlex_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

thf(sy_c_List_Olex_001_Eo,type,
    lex_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

thf(sy_c_List_Olex_001t__Int__Oint,type,
    lex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

thf(sy_c_List_Olex_001t__Nat__Onat,type,
    lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olex_001t__Real__Oreal,type,
    lex_real: set_Pr6218003697084177305l_real > set_Pr611352295856513593t_real ).

thf(sy_c_List_Olex_001t__Uint32__Ouint32,type,
    lex_uint32: set_Pr1773385645901665561uint32 > set_Pr2258164808687509945uint32 ).

thf(sy_c_List_Olex_001t__VEBT____BuildupMemImp__OVEBTi,type,
    lex_VEBT_VEBTi: set_Pr2227491710730465451_VEBTi > set_Pr4797120415438012619_VEBTi ).

thf(sy_c_List_Olex_001t__VEBT____Definitions__OVEBT,type,
    lex_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001_Eo,type,
    linord3142498349692569832_set_o: set_o > list_o ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Code____Numeral__Ointeger,type,
    linord2324613341767563021nteger: set_Code_integer > list_Code_integer ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Int__Oint,type,
    linord2612477271533052124et_int: set_int > list_int ).

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

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Num__Onum,type,
    linord8395671565052656842et_num: set_num > list_num ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Rat__Orat,type,
    linord1979837681955606664et_rat: set_rat > list_rat ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Real__Oreal,type,
    linord4252657396651189596t_real: set_real > list_real ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__String__Oliteral,type,
    linord2913955441264437540iteral: set_literal > list_literal ).

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

thf(sy_c_List_Olist_OCons_001t__Code____Numeral__Ointeger,type,
    cons_Code_integer: code_integer > list_Code_integer > list_Code_integer ).

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_OCons_001t__Num__Onum,type,
    cons_num: num > list_num > list_num ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    cons_P8766293264717362397od_o_o: product_prod_o_o > list_P4002435161011370285od_o_o > list_P4002435161011370285od_o_o ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    cons_P1455986808126089405_o_int: product_prod_o_int > list_P3795440434834930179_o_int > list_P3795440434834930179_o_int ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    cons_P5633837827635286113_o_nat: product_prod_o_nat > list_P6285523579766656935_o_nat > list_P6285523579766656935_o_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    cons_P7321330006258091179_int_o: product_prod_int_o > list_P5087981734274514673_int_o > list_P5087981734274514673_int_o ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    cons_P3334398858971670639nt_int: product_prod_int_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    cons_P7512249878480867347nt_nat: product_prod_int_nat > list_P8198026277950538467nt_nat > list_P8198026277950538467nt_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    cons_P9142372351690779143_nat_o: product_prod_nat_o > list_P7333126701944960589_nat_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    cons_P2335045147070616083at_int: product_prod_nat_int > list_P3521021558325789923at_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    cons_P3149448846263281007uint32: produc827990862158126777uint32 > list_P3069071885182933823uint32 > list_P3069071885182933823uint32 ).

thf(sy_c_List_Olist_OCons_001t__Rat__Orat,type,
    cons_rat: rat > list_rat > list_rat ).

thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
    cons_real: real > list_real > list_real ).

thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
    cons_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_Olist_OCons_001t__Uint32__Ouint32,type,
    cons_uint32: uint32 > list_uint32 > list_uint32 ).

thf(sy_c_List_Olist_OCons_001t__VEBT____BuildupMemImp__OVEBTi,type,
    cons_VEBT_VEBTi: vEBT_VEBTi > list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
    cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

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_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,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001_Eo,type,
    map_Pr7541730621154948341_o_o_o: ( product_prod_o_o > $o ) > list_P4002435161011370285od_o_o > list_o ).

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

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

thf(sy_c_List_Olist_Oset_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,
    set_Pr400311997697478754_nat_o: list_P7985473006766602707_nat_o > set_Pr4532377907799695533_nat_o ).

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

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

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

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

thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
    set_list_o2: list_list_o > set_list_o ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
    set_list_int2: list_list_int > set_list_int ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Real__Oreal_J,type,
    set_list_real2: list_list_real > set_list_real ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    set_list_VEBT_VEBT2: list_list_VEBT_VEBT > set_list_VEBT_VEBT ).

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

thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
    set_num2: list_num > set_num ).

thf(sy_c_List_Olist_Oset_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,
    set_Pr7458301993993767461et_nat: list_P362550909693114634et_nat > set_Pr8536935166611901872et_nat ).

thf(sy_c_List_Olist_Oset_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,
    set_Pr364071364088684201et_nat: list_P2321686559999237006et_nat > set_Pr3286484037609594932et_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    set_Pr3864771053587467565et_nat: list_P9062070895058802706et_nat > set_Pr3948176798113811640et_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    set_Pr8694291782656941196_int_o: list_P5087981734274514673_int_o > set_Pr903927857289325719_int_o ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    set_Pr6647972299459129970nt_nat: list_P8198026277950538467nt_nat > set_Pr3448869479623346877nt_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
    set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).

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

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
    set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    set_Pr2418681094996576974uint32: list_P3069071885182933823uint32 > set_Pr1773385645901665561uint32 ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).

thf(sy_c_List_Olist_Oset_001t__Rat__Orat,type,
    set_rat2: list_rat > set_rat ).

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

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

thf(sy_c_List_Olist_Oset_001t__String__Oliteral,type,
    set_literal2: list_literal > set_literal ).

thf(sy_c_List_Olist_Oset_001t__Uint32__Ouint32,type,
    set_uint322: list_uint32 > set_uint32 ).

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

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

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

thf(sy_c_List_Olist__update_001_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__update_001_Eo,type,
    list_update_o: list_o > nat > $o > list_o ).

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

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

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_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,
    list_u1913845194572276492et_nat: list_P362550909693114634et_nat > nat > produc2732055786443039994et_nat > list_P362550909693114634et_nat ).

thf(sy_c_List_Olist__update_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,
    list_u1866171116859985808et_nat: list_P2321686559999237006et_nat > nat > produc3925858234332021118et_nat > list_P2321686559999237006et_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    list_u1537252308907898773od_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o > list_P4002435161011370285od_o_o ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    list_u9060326803697358356et_nat: list_P9062070895058802706et_nat > nat > produc3658429121746597890et_nat > list_P9062070895058802706et_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    list_u2546161786664092711uint32: list_P3069071885182933823uint32 > nat > produc827990862158126777uint32 > list_P3069071885182933823uint32 ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    list_u30558089781959097_VEBTi: list_P785718909624839377_VEBTi > nat > produc3777764054643897931_VEBTi > list_P785718909624839377_VEBTi ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    list_u9044509791230035014T_VEBT: list_P5988454224134618948T_VEBT > nat > produc2810682830582626868T_VEBT > list_P5988454224134618948T_VEBT ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    list_u636002643828906794_VEBTi: list_P735349106241217576_VEBTi > nat > produc3625547720036274456_VEBTi > list_P735349106241217576_VEBTi ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    list_u6961636818849549845T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT > list_P7413028617227757229T_VEBT ).

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

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

thf(sy_c_List_Olist__update_001t__Uint32__Ouint32,type,
    list_update_uint32: list_uint32 > nat > uint32 > list_uint32 ).

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

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

thf(sy_c_List_Olistrel1_001_Eo,type,
    listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
    listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
    listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olistrel1_001t__Real__Oreal,type,
    listrel1_real: set_Pr6218003697084177305l_real > set_Pr611352295856513593t_real ).

thf(sy_c_List_Olistrel1_001t__Uint32__Ouint32,type,
    listrel1_uint32: set_Pr1773385645901665561uint32 > set_Pr2258164808687509945uint32 ).

thf(sy_c_List_Olistrel1_001t__VEBT____BuildupMemImp__OVEBTi,type,
    listrel1_VEBT_VEBTi: set_Pr2227491710730465451_VEBTi > set_Pr4797120415438012619_VEBTi ).

thf(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
    listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

thf(sy_c_List_Onth_001_Eo,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > nat > int ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
    nth_Pr731366597535767232t_real: list_P6863124054624500543t_real > nat > produc679980390762269497t_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr5092243222550848201_VEBTi: list_P4447955379000644808_VEBTi > nat > produc2062302194595185346_VEBTi ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__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,
    nth_Pr3244165891152107629_VEBTi: list_P2320588648998582380_VEBTi > nat > produc214224863196444774_VEBTi ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    nth_Pr6139322253122866368uint32: list_P3069071885182933823uint32 > nat > produc827990862158126777uint32 ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
    nth_Pr3433448822664029129i_real: list_P8536626330812492744i_real > nat > produc6680258955013199682i_real ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Rat__Orat,type,
    nth_rat: list_rat > nat > rat ).

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

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

thf(sy_c_List_Onth_001t__Uint32__Ouint32,type,
    nth_uint32: list_uint32 > nat > uint32 ).

thf(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_List_Oremove1_001_Eo,type,
    remove1_o: $o > list_o > list_o ).

thf(sy_c_List_Oremove1_001t__Code____Numeral__Ointeger,type,
    remove1_Code_integer: code_integer > list_Code_integer > list_Code_integer ).

thf(sy_c_List_Oremove1_001t__Int__Oint,type,
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thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
    remove1_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oremove1_001t__Num__Onum,type,
    remove1_num: num > list_num > list_num ).

thf(sy_c_List_Oremove1_001t__Rat__Orat,type,
    remove1_rat: rat > list_rat > list_rat ).

thf(sy_c_List_Oremove1_001t__Real__Oreal,type,
    remove1_real: real > list_real > list_real ).

thf(sy_c_List_Oremove1_001t__VEBT____Definitions__OVEBT,type,
    remove1_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_OremoveAll_001_Eo,type,
    removeAll_o: $o > list_o > list_o ).

thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
    removeAll_int: int > list_int > list_int ).

thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
    removeAll_nat: nat > list_nat > list_nat ).

thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
    removeAll_real: real > list_real > list_real ).

thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
    removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_OremoveAll_001t__VEBT____Definitions__OVEBT,type,
    removeAll_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

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

thf(sy_c_List_Orotate1_001_Eo,type,
    rotate1_o: list_o > list_o ).

thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

thf(sy_c_List_Orotate1_001t__Real__Oreal,type,
    rotate1_real: list_real > list_real ).

thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
    rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Osorted__wrt_001_Eo,type,
    sorted_wrt_o: ( $o > $o > $o ) > list_o > $o ).

thf(sy_c_List_Osorted__wrt_001t__Code____Numeral__Ointeger,type,
    sorted710888440204495920nteger: ( code_integer > code_integer > $o ) > list_Code_integer > $o ).

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

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

thf(sy_c_List_Osorted__wrt_001t__Num__Onum,type,
    sorted_wrt_num: ( num > num > $o ) > list_num > $o ).

thf(sy_c_List_Osorted__wrt_001t__Rat__Orat,type,
    sorted_wrt_rat: ( rat > rat > $o ) > list_rat > $o ).

thf(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
    sorted_wrt_real: ( real > real > $o ) > list_real > $o ).

thf(sy_c_List_Osorted__wrt_001t__String__Oliteral,type,
    sorted_wrt_literal: ( literal > literal > $o ) > list_literal > $o ).

thf(sy_c_List_Osorted__wrt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    sorted9206477368072086664_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi > $o ) > list_VEBT_VEBTi > $o ).

thf(sy_c_List_Osorted__wrt_001t__VEBT____Definitions__OVEBT,type,
    sorted_wrt_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > list_VEBT_VEBT > $o ).

thf(sy_c_List_Osubseqs_001_Eo,type,
    subseqs_o: list_o > list_list_o ).

thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
    subseqs_int: list_int > list_list_int ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

thf(sy_c_List_Osubseqs_001t__Real__Oreal,type,
    subseqs_real: list_real > list_list_real ).

thf(sy_c_List_Osubseqs_001t__VEBT____Definitions__OVEBT,type,
    subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).

thf(sy_c_List_Otake_001_Eo,type,
    take_o: nat > list_o > list_o ).

thf(sy_c_List_Otake_001t__Int__Oint,type,
    take_int: nat > list_int > list_int ).

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

thf(sy_c_List_Otake_001t__Num__Onum,type,
    take_num: nat > list_num > list_num ).

thf(sy_c_List_Otake_001t__Rat__Orat,type,
    take_rat: nat > list_rat > list_rat ).

thf(sy_c_List_Otake_001t__Real__Oreal,type,
    take_real: nat > list_real > list_real ).

thf(sy_c_List_Otake_001t__Set__Oset_It__Nat__Onat_J,type,
    take_set_nat: nat > list_set_nat > list_set_nat ).

thf(sy_c_List_Otake_001t__Uint32__Ouint32,type,
    take_uint32: nat > list_uint32 > list_uint32 ).

thf(sy_c_List_Otake_001t__VEBT____BuildupMemImp__OVEBTi,type,
    take_VEBT_VEBTi: nat > list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: nat > list_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_List_Ozip_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,
    zip_Pr8136144321567152340et_nat: list_P7985473006766602707_nat_o > list_P2321686559999237006et_nat > list_P362550909693114634et_nat ).

thf(sy_c_List_Ozip_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,
    zip_Pr7134870689397686104et_nat: list_P7985473006766602707_nat_o > list_P9062070895058802706et_nat > list_P2321686559999237006et_nat ).

thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
    zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
    zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
    zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
    zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).

thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
    zip_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
    zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
    zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
    zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    zip_int_VEBT_VEBTi: list_int > list_VEBT_VEBTi > list_P4447955379000644808_VEBTi ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
    zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Int__Oint,type,
    zip_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
    zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
    zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
    zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
    zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).

thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).

thf(sy_c_List_Ozip_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    zip_uint32_uint32: list_uint32 > list_uint32 > list_P3069071885182933823uint32 ).

thf(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    zip_VEBT_VEBTi_o: list_VEBT_VEBTi > list_o > list_P8833571063612306856EBTi_o ).

thf(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    zip_VEBT_VEBTi_int: list_VEBT_VEBTi > list_int > list_P7392757774524453448Ti_int ).

thf(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    zip_VEBT_VEBTi_real: list_VEBT_VEBTi > list_real > list_P8536626330812492744i_real ).

thf(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    zip_VE793581609497812771_VEBTi: list_VEBT_VEBTi > list_VEBT_VEBTi > list_P785718909624839377_VEBTi ).

thf(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    zip_VE7413257051550508102T_VEBT: list_VEBT_VEBTi > list_VEBT_VEBT > list_P5988454224134618948T_VEBT ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    zip_VE6444338338598820466_VEBTi: list_VEBT_VEBT > list_VEBT_VEBTi > list_P735349106241217576_VEBTi ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(sy_c_Misc_Obijective_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,
    biject1468766312547416318et_nat: set_Pr8536935166611901872et_nat > $o ).

thf(sy_c_Misc_Obijective_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,
    biject2615096655818420098et_nat: set_Pr3286484037609594932et_nat > $o ).

thf(sy_c_Misc_Obijective_001t__Int__Oint_001t__Int__Oint,type,
    bijective_int_int: set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Misc_Obijective_001t__Nat__Onat_001t__Nat__Onat,type,
    bijective_nat_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Misc_Obijective_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    biject2923219584331114343uint32: set_Pr1773385645901665561uint32 > $o ).

thf(sy_c_Misc_Odflt__None__set_001_Eo,type,
    dflt_None_set_o: set_o > option_set_o ).

thf(sy_c_Misc_Odflt__None__set_001t__Int__Oint,type,
    dflt_None_set_int: set_int > option_set_int ).

thf(sy_c_Misc_Odflt__None__set_001t__Nat__Onat,type,
    dflt_None_set_nat: set_nat > option_set_nat ).

thf(sy_c_Misc_Omergesort__remdups_001_Eo,type,
    mergesort_remdups_o: list_o > list_o ).

thf(sy_c_Misc_Omergesort__remdups_001t__Int__Oint,type,
    merges1654122896337280797ps_int: list_int > list_int ).

thf(sy_c_Misc_Omergesort__remdups_001t__Nat__Onat,type,
    merges1656613366846331073ps_nat: list_nat > list_nat ).

thf(sy_c_Misc_Omergesort__remdups_001t__Num__Onum,type,
    merges7437317189856885515ps_num: list_num > list_num ).

thf(sy_c_Misc_Omergesort__remdups_001t__Rat__Orat,type,
    merges1021483306759835337ps_rat: list_rat > list_rat ).

thf(sy_c_Misc_Omergesort__remdups_001t__Real__Oreal,type,
    merges7559785487730971421s_real: list_real > list_real ).

thf(sy_c_Misc_Oslice_001_Eo,type,
    slice_o: nat > nat > list_o > list_o ).

thf(sy_c_Misc_Oslice_001t__Int__Oint,type,
    slice_int: nat > nat > list_int > list_int ).

thf(sy_c_Misc_Oslice_001t__Nat__Onat,type,
    slice_nat: nat > nat > list_nat > list_nat ).

thf(sy_c_Misc_Oslice_001t__Real__Oreal,type,
    slice_real: nat > nat > list_real > list_real ).

thf(sy_c_Misc_Oslice_001t__VEBT____BuildupMemImp__OVEBTi,type,
    slice_VEBT_VEBTi: nat > nat > list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_Misc_Oslice_001t__VEBT____Definitions__OVEBT,type,
    slice_VEBT_VEBT: nat > nat > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_Misc_Othe__default_001t__Nat__Onat,type,
    the_default_nat: nat > option_nat > nat ).

thf(sy_c_Misc_Othe__default_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_de3812365490807259288at_nat: product_prod_nat_nat > option4927543243414619207at_nat > product_prod_nat_nat ).

thf(sy_c_Misc_Othe__default_001t__Set__Oset_I_Eo_J,type,
    the_default_set_o: set_o > option_set_o > set_o ).

thf(sy_c_Misc_Othe__default_001t__Set__Oset_It__Int__Oint_J,type,
    the_default_set_int: set_int > option_set_int > set_int ).

thf(sy_c_Misc_Othe__default_001t__Set__Oset_It__Nat__Onat_J,type,
    the_default_set_nat: set_nat > option_set_nat > set_nat ).

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

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

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

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_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
    size_s2710708370519433104list_o: list_list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
    size_s6660260683639930848t_real: list_list_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_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_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__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__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__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s3289364478449617953T_VEBT: list_P877281246627933069T_VEBT > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    size_s5035110155006384947T_real: list_P2623026923184700063T_real > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Rat__Orat_J,type,
    size_size_list_rat: list_rat > 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__String__Oliteral_J,type,
    size_s2501651207091587910iteral: list_literal > 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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____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__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Uint32__Ouint32,type,
    neg_nu965353292909893953uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__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_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_Osize__num,type,
    size_num: num > nat ).

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

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__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_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_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_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_It__Int__Oint_Mt__Int__Oint_J,type,
    none_P2377608414092835994nt_int: option4624381673175914239nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    none_P7441528354948028570uint32: option3972171592325465343uint32 ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_I_Eo_J,type,
    none_set_o: option_set_o ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Int__Oint_J,type,
    none_set_int: option_set_int ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

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

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

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

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_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_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__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    some_P6695178580830101022uint32: produc827990862158126777uint32 > option3972171592325465343uint32 ).

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_I_Eo_J,type,
    some_set_o: set_o > option_set_o ).

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_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o1383228350324149268at_nat: option_nat > ( product_prod_nat_nat > option_nat ) > option4927543243414619207at_nat > option_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o2442805151034396888at_nat: vEBT_VEBT > ( product_prod_nat_nat > vEBT_VEBT ) > option4927543243414619207at_nat > vEBT_VEBT ).

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

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

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

thf(sy_c_Orderings_Obot__class_Obot_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,
    bot_bo3790638025767943357_nat_o: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_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,
    bot_bo5580076615179976505_nat_o: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
    bot_bot_o_o: $o > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    bot_bot_complex_o: complex > $o ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
    bot_bot_int_o: int > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_list_nat_o: list_nat > $o ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
    bot_bot_nat_o: nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
    bot_bot_real_o: real > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_M_Eo_J_J,type,
    bot_bo8112279147790955290nt32_o: uint32 > uint32 > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    bot_bot_VEBT_VEBT_o: vEBT_VEBT > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
    bot_bot_o: $o ).

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_I_Eo_J,type,
    bot_bot_set_o: set_o ).

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__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    bot_bot_set_list_nat: set_list_nat ).

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_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,
    bot_bo5635537948650799172et_nat: set_Pr8536935166611901872et_nat ).

thf(sy_c_Orderings_Obot__class_Obot_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,
    bot_bo1481135142794719944et_nat: set_Pr3286484037609594932et_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    bot_bo7073875226086086771od_o_o: set_Product_prod_o_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J,type,
    bot_bo8438649754162204037uint32: set_Pr1773385645901665561uint32 ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    bot_bo5088076668136028147T_VEBT: set_Pr6192946355708809607T_VEBT ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__String__Oliteral_J,type,
    bot_bot_set_literal: set_literal ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    bot_bot_set_uint32: set_uint32 ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    bot_bo8982466882572371071_VEBTi: set_VEBT_VEBTi ).

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_OLeast_001t__Real__Oreal,type,
    ord_Least_real: ( real > $o ) > real ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Assertions__Oassn,type,
    ord_less_assn: assn > assn > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__String__Oliteral_J,type,
    ord_less_set_literal: set_literal > set_literal > $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__String__Ochar,type,
    ord_less_char: char > char > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__String__Oliteral,type,
    ord_less_literal: literal > literal > $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__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_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__Real__Oreal_M_Eo_J,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_M_Eo_J_J,type,
    ord_le5999336846926846542nt32_o: ( uint32 > uint32 > $o ) > ( uint32 > uint32 > $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_001_Eo,type,
    ord_less_eq_o: $o > $o > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__List__Olist_It__Nat__Onat_J_J,type,
    ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $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__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_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J,type,
    ord_le6429528607962791097uint32: set_Pr1773385645901665561uint32 > set_Pr1773385645901665561uint32 > $o ).

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__String__Oliteral_J,type,
    ord_le7307670543136651348iteral: set_literal > set_literal > $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__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_001_Eo,type,
    ord_max_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Assertions__Oassn,type,
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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_I_Eo_J,type,
    ord_max_set_o: set_o > set_o > set_o ).

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

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

thf(sy_c_Orderings_Oord__class_Omin_001t__Code____Numeral__Ointeger,type,
    ord_min_Code_integer: code_integer > code_integer > code_integer ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: int > int > int ).

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_Oordering__top_001t__Assertions__Oassn,type,
    ordering_top_assn: ( assn > assn > $o ) > ( assn > assn > $o ) > assn > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_I_Eo_J,type,
    ordering_top_set_o: ( set_o > set_o > $o ) > ( set_o > set_o > $o ) > set_o > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Int__Oint_J,type,
    ordering_top_set_int: ( set_int > set_int > $o ) > ( set_int > set_int > $o ) > set_int > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
    ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__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__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
    top_top_set_literal: set_literal ).

thf(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
    power_power_assn: assn > nat > assn ).

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

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

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

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

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

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

thf(sy_c_Power_Opower__class_Opower_001t__Uint32__Ouint32,type,
    power_power_uint32: uint32 > nat > uint32 ).

thf(sy_c_Product__Type_OPair_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,
    produc2245416461498447860et_nat: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat ).

thf(sy_c_Product__Type_OPair_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,
    produc5001842942810119800et_nat: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: $o > int > product_prod_o_int ).

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

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001_Eo,type,
    product_Pair_int_o: int > $o > product_prod_int_o ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
    product_Pair_int_nat: int > nat > product_prod_int_nat ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
    produc801115645435158769t_real: int > real > produc679980390762269497t_real ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc4497823428076633978_VEBTi: int > vEBT_VEBTi > produc2062302194595185346_VEBTi ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
    produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
    produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Real__Oreal_J_001t__List__Olist_It__Real__Oreal_J,type,
    produc1408950526243324945t_real: list_real > list_real > produc478978216448986841t_real ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Uint32__Ouint32_J_001t__List__Olist_It__Uint32__Ouint32_J,type,
    produc7487160679990061969uint32: list_uint32 > list_uint32 > produc823665806539077721uint32 ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    produc4384243565435462691_VEBTi: list_VEBT_VEBTi > list_VEBT_VEBTi > produc3089554586268799851_VEBTi ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

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

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
    produc7837566107596912789t_real: nat > real > produc7716430852924023517t_real ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc2649746096677893406_VEBTi: nat > vEBT_VEBTi > produc214224863196444774_VEBTi ).

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

thf(sy_c_Product__Type_OPair_001t__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_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,
    produc1940133919992309389et_nat: produc2732055786443039994et_nat > produc2732055786443039994et_nat > produc5657529347773406293et_nat ).

thf(sy_c_Product__Type_OPair_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_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc8599840265553166229et_nat: produc3925858234332021118et_nat > produc3925858234332021118et_nat > produc6830853553727218525et_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    produc6884863695460953815uint32: produc827990862158126777uint32 > produc827990862158126777uint32 > produc8822557026176459367uint32 ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

thf(sy_c_Product__Type_OPair_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    produc8194178580519725514EBTi_o: vEBT_VEBTi > $o > produc5014006835512566296EBTi_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    produc7190175284220460154Ti_int: vEBT_VEBTi > int > produc8927069083091599426Ti_int ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc8457151488442208762i_real: vEBT_VEBTi > real > produc6680258955013199682i_real ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc436343169921013763_VEBTi: vEBT_VEBTi > vEBT_VEBTi > produc3777764054643897931_VEBTi ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc7053807326796202854T_VEBT: vEBT_VEBTi > vEBT_VEBT > produc2810682830582626868T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc6084888613844515218_VEBTi: vEBT_VEBT > vEBT_VEBTi > produc3625547720036274456_VEBTi ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_Eo_001_Eo_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_4702325430467532143et_nat: ( set_int > set_nat ) > set_set_int > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
    image_set_nat_int: ( set_nat > int ) > set_set_nat > set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
    image_3739036796817536367et_int: ( set_nat > set_int ) > set_set_nat > set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
    image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    image_1661326939266726661T_VEBT: ( set_VEBT_VEBT > set_VEBT_VEBT ) > set_set_VEBT_VEBT > set_set_VEBT_VEBT ).

thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Code____Numeral__Ointeger,type,
    image_4442872163159017964nteger: ( literal > code_integer ) > set_literal > set_Code_integer ).

thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Complex__Ocomplex,type,
    image_5274195009022015549omplex: ( literal > complex ) > set_literal > set_complex ).

thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Nat__Onat,type,
    image_literal_nat: ( literal > nat ) > set_literal > set_nat ).

thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__String__Oliteral,type,
    image_8195128725298311301iteral: ( literal > literal ) > set_literal > set_literal ).

thf(sy_c_Set_Oimage_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    image_uint32_uint32: ( uint32 > uint32 ) > set_uint32 > set_uint32 ).

thf(sy_c_Set_Oimage_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    image_VEBT_VEBTi_int: ( vEBT_VEBTi > int ) > set_VEBT_VEBTi > set_int ).

thf(sy_c_Set_Oimage_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    image_VEBT_VEBTi_nat: ( vEBT_VEBTi > nat ) > set_VEBT_VEBTi > set_nat ).

thf(sy_c_Set_Oimage_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    image_6202559892754154600i_real: ( vEBT_VEBTi > real ) > set_VEBT_VEBTi > set_real ).

thf(sy_c_Set_Oimage_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    image_7547481670047419768T_VEBT: ( vEBT_VEBTi > vEBT_VEBT ) > set_VEBT_VEBTi > set_VEBT_VEBT ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    image_VEBT_VEBT_o: ( vEBT_VEBT > $o ) > set_VEBT_VEBT > set_o ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Code____Numeral__Ointeger,type,
    image_2092689629700589388nteger: ( vEBT_VEBT > code_integer ) > set_VEBT_VEBT > set_Code_integer ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    image_3793382806556112285omplex: ( vEBT_VEBT > complex ) > set_VEBT_VEBT > set_complex ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    image_VEBT_VEBT_int: ( vEBT_VEBT > int ) > set_VEBT_VEBT > set_int ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    image_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > set_nat ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    image_VEBT_VEBT_real: ( vEBT_VEBT > real ) > set_VEBT_VEBT > set_real ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    image_3375948659692109573T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > set_VEBT_VEBT > set_VEBT_VEBT ).

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

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

thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
    insert_complex: complex > set_complex > set_complex ).

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

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: list_nat > set_list_nat > set_list_nat ).

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

thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
    insert_num: num > set_num > set_num ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    insert6201435330877294327od_o_o: product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).

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

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    insert4454361187789264009uint32: produc827990862158126777uint32 > set_Pr1773385645901665561uint32 > set_Pr1773385645901665561uint32 ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    insert494605675473494903T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > set_Pr6192946355708809607T_VEBT ).

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

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

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

thf(sy_c_Set_Oinsert_001t__String__Oliteral,type,
    insert_literal: literal > set_literal > set_literal ).

thf(sy_c_Set_Oinsert_001t__Uint32__Ouint32,type,
    insert_uint32: uint32 > set_uint32 > set_uint32 ).

thf(sy_c_Set_Oinsert_001t__VEBT____BuildupMemImp__OVEBTi,type,
    insert_VEBT_VEBTi: vEBT_VEBTi > set_VEBT_VEBTi > set_VEBT_VEBTi ).

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

thf(sy_c_Set_Ois__singleton_001_Eo,type,
    is_singleton_o: set_o > $o ).

thf(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
    is_singleton_complex: set_complex > $o ).

thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
    is_singleton_int: set_int > $o ).

thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
    is_sin2641923865335537900st_nat: set_list_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
    is_singleton_nat: set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
    is_singleton_real: set_real > $o ).

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
    is_singleton_set_nat: set_set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__String__Oliteral,type,
    is_singleton_literal: set_literal > $o ).

thf(sy_c_Set_Ois__singleton_001t__VEBT____Definitions__OVEBT,type,
    is_sin24926331636114728T_VEBT: set_VEBT_VEBT > $o ).

thf(sy_c_Set_Oremove_001_Eo,type,
    remove_o: $o > set_o > set_o ).

thf(sy_c_Set_Oremove_001t__Int__Oint,type,
    remove_int: int > set_int > set_int ).

thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
    remove_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oremove_001t__Real__Oreal,type,
    remove_real: real > set_real > set_real ).

thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
    remove_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oremove_001t__VEBT____Definitions__OVEBT,type,
    remove_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Set_Othe__elem_001_Eo,type,
    the_elem_o: set_o > $o ).

thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
    the_elem_int: set_int > int ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
    the_elem_set_nat: set_set_nat > set_nat ).

thf(sy_c_Set_Othe__elem_001t__VEBT____Definitions__OVEBT,type,
    the_elem_VEBT_VEBT: set_VEBT_VEBT > vEBT_VEBT ).

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
    set_or189985376899183464nteger: code_integer > code_integer > set_Code_integer ).

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
    set_or7139685690850216873Than_o: $o > $o > set_o ).

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
    set_or1222409239386451017an_num: num > num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
    set_or4029947393144176647an_rat: rat > rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
    set_or66887138388493659n_real: real > real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Int__Oint_J,type,
    set_or8585797421378605585et_int: set_int > set_int > set_set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

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

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
    set_or2715278749043346189nteger: code_integer > code_integer > set_Code_integer ).

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

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Code____Numeral__Ointeger,type,
    set_or4266950643985792945nteger: code_integer > code_integer > set_Code_integer ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: int > int > int ).

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

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__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__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > nat > $o ).

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

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

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

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

thf(sy_c_Time__Reasoning_Otime_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_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

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

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

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

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

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

thf(sy_c_Transcendental_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__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_Transitive__Closure_Otrancl_001_Eo,type,
    transitive_trancl_o: set_Product_prod_o_o > set_Product_prod_o_o ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Int__Oint,type,
    transi6261509568448316235cl_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

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

thf(sy_c_Transitive__Closure_Otrancl_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,
    transi5221092739591632921et_nat: set_Pr3444600963470892981et_nat > set_Pr3444600963470892981et_nat ).

thf(sy_c_Transitive__Closure_Otrancl_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,
    transi3145040225084697757et_nat: set_Pr7928877670098842301et_nat > set_Pr7928877670098842301et_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    transi6288783178788033498nt_int: set_Pr2560585780119916871nt_int > set_Pr2560585780119916871nt_int ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    transi243908449541399842at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    transi5456044692924788698uint32: set_Pr3773659940955823943uint32 > set_Pr3773659940955823943uint32 ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Real__Oreal,type,
    transi1789104906590519371l_real: set_Pr6218003697084177305l_real > set_Pr6218003697084177305l_real ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Uint32__Ouint32,type,
    transi3114468042090999947uint32: set_Pr1773385645901665561uint32 > set_Pr1773385645901665561uint32 ).

thf(sy_c_Transitive__Closure_Otrancl_001t__VEBT____BuildupMemImp__OVEBTi,type,
    transi2803566869205510612_VEBTi: set_Pr2227491710730465451_VEBTi > set_Pr2227491710730465451_VEBTi ).

thf(sy_c_Transitive__Closure_Otrancl_001t__VEBT____Definitions__OVEBT,type,
    transi8906537157094044885T_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr6192946355708809607T_VEBT ).

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

thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    type_d8366093980585677751er_int: ( code_integer > int ) > ( int > code_integer ) > set_int > $o ).

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

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_ORep__uint32,type,
    rep_uint32: uint32 > word_N3645301735248828278l_num1 ).

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_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

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_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__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_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__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

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__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

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

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

thf(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: nat > set_nat > vEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6286945158656146733_VEBTi: set_nat > ( $o > vEBT_VEBTi > assn ) > list_o > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1319876754960170684T_VEBT: set_nat > ( $o > vEBT_VEBT > assn ) > list_o > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001_Eo,type,
    vEBT_L3563379889750563018_int_o: set_nat > ( int > $o > assn ) > list_int > list_o > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_L8888932350013902202nt_int: set_nat > ( int > int > assn ) > list_int > list_int > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_L8891422820522952478nt_nat: set_nat > ( int > nat > assn ) > list_int > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L7077748017936769786t_real: set_nat > ( int > real > assn ) > list_int > list_real > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L114188773329725699_VEBTi: set_nat > ( int > vEBT_VEBTi > assn ) > list_int > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2018189785592951398T_VEBT: set_nat > ( int > vEBT_VEBT > assn ) > list_int > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7489483478785760935_VEBTi: set_nat > ( nat > vEBT_VEBTi > assn ) > list_nat > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8511957252848910786T_VEBT: set_nat > ( nat > vEBT_VEBT > assn ) > list_nat > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7851252805511451907_VEBTi: set_nat > ( real > vEBT_VEBTi > assn ) > list_real > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3095048238742455910T_VEBT: set_nat > ( real > vEBT_VEBT > assn ) > list_real > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    vEBT_L3328983362619735041EBTi_o: set_nat > ( vEBT_VEBTi > $o > assn ) > list_VEBT_VEBTi > list_o > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L2806540629473551875Ti_int: set_nat > ( vEBT_VEBTi > int > assn ) > list_VEBT_VEBTi > list_int > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L2809031099982602151Ti_nat: set_nat > ( vEBT_VEBTi > nat > assn ) > list_VEBT_VEBTi > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    vEBT_L7728200936804140803i_real: set_nat > ( vEBT_VEBTi > real > assn ) > list_VEBT_VEBTi > list_real > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L886525131989349516_VEBTi: set_nat > ( vEBT_VEBTi > vEBT_VEBTi > assn ) > list_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2497118539674116125T_VEBT: set_nat > ( vEBT_VEBTi > vEBT_VEBT > assn ) > list_VEBT_VEBTi > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8650695023172932196BT_nat: set_nat > ( vEBT_VEBT > nat > assn ) > list_VEBT_VEBT > list_nat > assn ).

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_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3204528365124325536T_VEBT: set_nat > ( vEBT_VEBT > vEBT_VEBT > assn ) > list_VEBT_VEBT > list_VEBT_VEBT > 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_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1750719106661372127T_VEBT: ( $o > vEBT_VEBT > assn ) > list_o > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_L74593716426352029nt_int: ( int > int > assn ) > list_int > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_L77084186935402305nt_nat: ( int > nat > assn ) > list_int > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6235239671944049190_VEBTi: ( int > vEBT_VEBTi > assn ) > list_int > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1664421287176695555T_VEBT: ( int > vEBT_VEBT > assn ) > list_int > list_VEBT_VEBT > 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__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____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L8927591528087875366Ti_int: ( vEBT_VEBTi > int > assn ) > list_VEBT_VEBTi > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L8930081998596925642Ti_nat: ( vEBT_VEBTi > nat > assn ) > list_VEBT_VEBTi > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1891944875198410415_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi > assn ) > list_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L7265847600308530106T_VEBT: ( vEBT_VEBTi > vEBT_VEBT > assn ) > list_VEBT_VEBTi > 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__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__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_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_VEBT__Uniqueness_OVEBT__internal_OperInsTrans,type,
    vEBT_V6289311342943941716sTrans: vEBT_VEBT > vEBT_VEBT > $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__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,
    accp_P1862375125659990705et_nat: ( produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ) > produc2732055786443039994et_nat > $o ).

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

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

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

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

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

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

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

thf(sy_c_member_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,
    member6576561426505652726_nat_o: ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > $o ).

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

thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $o ).

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

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

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

thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: list_real > set_list_real > $o ).

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_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,
    member6124377750444531601et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > $o ).

thf(sy_c_member_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,
    member1996754912294343701et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $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_M_Eo_J,type,
    member4489920277610959864_int_o: product_prod_int_o > set_Pr903927857289325719_int_o > $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__Int__Oint_Mt__Nat__Onat_J,type,
    member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
    member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
    member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Real__Oreal_J_Mt__List__Olist_It__Real__Oreal_J_J,type,
    member6584958104391596930t_real: produc478978216448986841t_real > set_Pr611352295856513593t_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Uint32__Ouint32_J_Mt__List__Olist_It__Uint32__Ouint32_J_J,type,
    member2333554998283850498uint32: produc823665806539077721uint32 > set_Pr2258164808687509945uint32 > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    member4173000155140927252_VEBTi: produc3089554586268799851_VEBTi > set_Pr4797120415438012619_VEBTi > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $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__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_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_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,
    member6341495586645257982et_nat: produc5657529347773406293et_nat > set_Pr3444600963470892981et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_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_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,
    member4763271486408492550et_nat: produc6830853553727218525et_nat > set_Pr7928877670098842301et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_Mt__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J,type,
    member4724532482705224080uint32: produc8822557026176459367uint32 > set_Pr3773659940955823943uint32 > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
    member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $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__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    member660371905731732212_VEBTi: produc3777764054643897931_VEBTi > set_Pr2227491710730465451_VEBTi > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $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_I_Eo_J,type,
    member_set_o: set_o > set_set_o > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
    member_set_real: set_real > set_set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: set_set_nat > set_set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    member_set_VEBT_VEBT: set_VEBT_VEBT > set_set_VEBT_VEBT > $o ).

thf(sy_c_member_001t__String__Oliteral,type,
    member_literal: literal > set_literal > $o ).

thf(sy_c_member_001t__Uint32__Ouint32,type,
    member_uint32: uint32 > set_uint32 > $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_a,type,
    a: heap_e7401611519738050253t_unit ).

thf(sy_v_b,type,
    b: set_nat ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_r,type,
    r: vEBT_VEBTi ).

thf(sy_v_s,type,
    s: set_nat ).

thf(sy_v_t,type,
    t: vEBT_VEBT ).

thf(sy_v_x,type,
    x: nat ).

thf(sy_v_xa,type,
    xa: nat ).

% Relevant facts (9099)
thf(fact_0_invar__vebt__delete,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X ) @ N ) ) ).

% invar_vebt_delete
thf(fact_1_set__vebt__equal,axiom,
    ! [T_1: vEBT_VEBT,N: nat,T_2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T_1 @ N )
     => ( ( vEBT_invar_vebt @ T_2 @ N )
       => ( ( T_1 = T_2 )
          = ( ( vEBT_set_vebt @ T_1 )
            = ( vEBT_set_vebt @ T_2 ) ) ) ) ) ).

% set_vebt_equal
thf(fact_2_unique__tree,axiom,
    ! [T: vEBT_VEBT,N: nat,S: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_invar_vebt @ S @ N )
       => ( ( ( vEBT_set_vebt @ T )
            = ( vEBT_set_vebt @ S ) )
         => ( S = T ) ) ) ) ).

% unique_tree
thf(fact_3_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T @ X ) @ N ) ) ).

% insert'_pres_valid
thf(fact_4_perIT__concat,axiom,
    ! [S: vEBT_VEBT,T: vEBT_VEBT,U: vEBT_VEBT] :
      ( ( vEBT_V6289311342943941716sTrans @ S @ T )
     => ( ( vEBT_V6289311342943941716sTrans @ T @ U )
       => ( vEBT_V6289311342943941716sTrans @ S @ U ) ) ) ).

% perIT_concat
thf(fact_5_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_6_uniquetree,axiom,
    ! [T: vEBT_VEBT,N: nat,S: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_invar_vebt @ S @ N )
       => ( ( ( vEBT_VEBT_set_vebt @ T )
            = ( vEBT_VEBT_set_vebt @ S ) )
         => ( S = T ) ) ) ) ).

% uniquetree
thf(fact_7_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_8_prod_Oinject,axiom,
    ! [X1: uint32,X2: uint32,Y1: uint32,Y2: uint32] :
      ( ( ( produc1400373151660368625uint32 @ X1 @ X2 )
        = ( produc1400373151660368625uint32 @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_9_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( ( product_Pair_int_int @ X1 @ X2 )
        = ( product_Pair_int_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ X1 @ X2 )
        = ( produc5001842942810119800et_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ X1 @ X2 )
        = ( produc2245416461498447860et_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
    ! [A: uint32,B: uint32,A2: uint32,B2: uint32] :
      ( ( ( produc1400373151660368625uint32 @ A @ B )
        = ( produc1400373151660368625uint32 @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_18_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_19_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_20_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_21_set__vebt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% set_vebt_member
thf(fact_22_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_23_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_24_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_25_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_26_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P @ ( produc5001842942810119800et_nat @ Q @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_27_prod__induct4,axiom,
    ! [P2: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: heap_e7401611519738050253t_unit,D2: set_nat] : ( P2 @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ ( produc7507926704131184380et_nat @ C @ D2 ) ) ) )
     => ( P2 @ X ) ) ).

% prod_induct4
thf(fact_28_prod__induct3,axiom,
    ! [P2: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: heap_e7401611519738050253t_unit,C: set_nat] : ( P2 @ ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_29_prod__induct3,axiom,
    ! [P2: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat] : ( P2 @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_30_prod__cases4,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: heap_e7401611519738050253t_unit,D2: set_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ ( produc7507926704131184380et_nat @ C @ D2 ) ) ) ) ).

% prod_cases4
thf(fact_31_prod__cases3,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: heap_e7401611519738050253t_unit,C: set_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_32_prod__cases3,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_33_Pair__inject,axiom,
    ! [A: uint32,B: uint32,A2: uint32,B2: uint32] :
      ( ( ( produc1400373151660368625uint32 @ A @ B )
        = ( produc1400373151660368625uint32 @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_34_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_35_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_36_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_37_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_38_prod__cases,axiom,
    ! [P2: produc827990862158126777uint32 > $o,P3: produc827990862158126777uint32] :
      ( ! [A3: uint32,B3: uint32] : ( P2 @ ( produc1400373151660368625uint32 @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_39_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_40_prod__cases,axiom,
    ! [P2: product_prod_int_int > $o,P3: product_prod_int_int] :
      ( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_41_prod__cases,axiom,
    ! [P2: produc3925858234332021118et_nat > $o,P3: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] : ( P2 @ ( produc5001842942810119800et_nat @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_42_prod__cases,axiom,
    ! [P2: produc2732055786443039994et_nat > $o,P3: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] : ( P2 @ ( produc2245416461498447860et_nat @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_43_surj__pair,axiom,
    ! [P3: produc827990862158126777uint32] :
    ? [X3: uint32,Y3: uint32] :
      ( P3
      = ( produc1400373151660368625uint32 @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_44_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P3
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_45_surj__pair,axiom,
    ! [P3: product_prod_int_int] :
    ? [X3: int,Y3: int] :
      ( P3
      = ( product_Pair_int_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_46_surj__pair,axiom,
    ! [P3: produc3925858234332021118et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
      ( P3
      = ( produc5001842942810119800et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_47_surj__pair,axiom,
    ! [P3: produc2732055786443039994et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
      ( P3
      = ( produc2245416461498447860et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_48_old_Oprod_Oexhaust,axiom,
    ! [Y: produc827990862158126777uint32] :
      ~ ! [A3: uint32,B3: uint32] :
          ( Y
         != ( produc1400373151660368625uint32 @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_49_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_50_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] :
          ( Y
         != ( product_Pair_int_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_51_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_53_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H @ As ) ) ).

% one_assn_raw.cases
thf(fact_54_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_55_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
    ! [A: real,P2: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
    ! [A: complex,P2: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
    ! [A: int,P2: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_62_Collect__mem__eq,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
    ! [A4: set_real] :
      ( ( collect_real
        @ ^ [X4: real] : ( member_real @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
    ! [A4: set_complex] :
      ( ( collect_complex
        @ ^ [X4: complex] : ( member_complex @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
    ! [A4: set_int] :
      ( ( collect_int
        @ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_69_Collect__cong,axiom,
    ! [P2: complex > $o,Q2: complex > $o] :
      ( ! [X3: complex] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_complex @ P2 )
        = ( collect_complex @ Q2 ) ) ) ).

% Collect_cong
thf(fact_70_Collect__cong,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_list_nat @ P2 )
        = ( collect_list_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_71_Collect__cong,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_set_nat @ P2 )
        = ( collect_set_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_72_Collect__cong,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_73_Collect__cong,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_int @ P2 )
        = ( collect_int @ Q2 ) ) ) ).

% Collect_cong
thf(fact_74_VEBT__internal_OperInsTrans_Ointros_I1_J,axiom,
    ! [T: vEBT_VEBT] : ( vEBT_V6289311342943941716sTrans @ T @ T ) ).

% VEBT_internal.perInsTrans.intros(1)
thf(fact_75_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_76_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_77_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_78_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_79_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_80_ssubst__Pair__rhs,axiom,
    ! [R: uint32,S: uint32,R2: set_Pr1773385645901665561uint32,S2: uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_81_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_82_ssubst__Pair__rhs,axiom,
    ! [R: int,S: int,R2: set_Pr958786334691620121nt_int,S2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_83_ssubst__Pair__rhs,axiom,
    ! [R: produc3658429121746597890et_nat > $o,S: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: produc3658429121746597890et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_84_ssubst__Pair__rhs,axiom,
    ! [R: produc3658429121746597890et_nat > $o,S: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: produc3925858234332021118et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_85_bex2I,axiom,
    ! [A: uint32,B: uint32,S3: set_Pr1773385645901665561uint32,P2: uint32 > uint32 > $o] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ S3 )
     => ( ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ S3 )
         => ( P2 @ A @ B ) )
       => ? [A3: uint32,B3: uint32] :
            ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A3 @ B3 ) @ S3 )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_86_bex2I,axiom,
    ! [A: nat,B: nat,S3: set_Pr1261947904930325089at_nat,P2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S3 )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S3 )
         => ( P2 @ A @ B ) )
       => ? [A3: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S3 )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_87_bex2I,axiom,
    ! [A: int,B: int,S3: set_Pr958786334691620121nt_int,P2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S3 )
     => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S3 )
         => ( P2 @ A @ B ) )
       => ? [A3: int,B3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ S3 )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_88_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S3: set_Pr3286484037609594932et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S3 )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S3 )
         => ( P2 @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A3 @ B3 ) @ S3 )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_89_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S3: set_Pr8536935166611901872et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S3 )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S3 )
         => ( P2 @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A3 @ B3 ) @ S3 )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_90_set__vebt__minNull,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ T )
        = ( ( vEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% set_vebt_minNull
thf(fact_91_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_92_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% deg_not_0
thf(fact_93_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S4 ) ) ) ).

% deg_SUcn_Node
thf(fact_94_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_95_divides__aux__eq,axiom,
    ! [Q3: nat,R: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R ) )
      = ( R = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_96_divides__aux__eq,axiom,
    ! [Q3: int,R: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R ) )
      = ( R = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_97_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_98_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X22 )
        = ( vEBT_Leaf @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% VEBT.inject(2)
thf(fact_99_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_100_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs )
              & ( ord_less_nat @ A @ X5 ) ) ) ) ).

% succ_none_empty
thf(fact_101_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs )
              & ( ord_less_nat @ X5 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_102_memb__imp__not__empty,axiom,
    ! [X: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ S3 )
     => ( S3 != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
thf(fact_103_memb__imp__not__empty,axiom,
    ! [X: real,S3: set_real] :
      ( ( member_real @ X @ S3 )
     => ( S3 != bot_bot_set_real ) ) ).

% memb_imp_not_empty
thf(fact_104_memb__imp__not__empty,axiom,
    ! [X: set_nat,S3: set_set_nat] :
      ( ( member_set_nat @ X @ S3 )
     => ( S3 != bot_bot_set_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_105_memb__imp__not__empty,axiom,
    ! [X: nat,S3: set_nat] :
      ( ( member_nat @ X @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_106_memb__imp__not__empty,axiom,
    ! [X: int,S3: set_int] :
      ( ( member_int @ X @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
thf(fact_107_memb__imp__not__empty,axiom,
    ! [X: $o,S3: set_o] :
      ( ( member_o @ X @ S3 )
     => ( S3 != bot_bot_set_o ) ) ).

% memb_imp_not_empty
thf(fact_108_set__notEmptyE,axiom,
    ! [S3: set_VEBT_VEBT] :
      ( ( S3 != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X3: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_109_set__notEmptyE,axiom,
    ! [S3: set_real] :
      ( ( S3 != bot_bot_set_real )
     => ~ ! [X3: real] :
            ~ ( member_real @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_110_set__notEmptyE,axiom,
    ! [S3: set_set_nat] :
      ( ( S3 != bot_bot_set_set_nat )
     => ~ ! [X3: set_nat] :
            ~ ( member_set_nat @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_111_set__notEmptyE,axiom,
    ! [S3: set_nat] :
      ( ( S3 != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_112_set__notEmptyE,axiom,
    ! [S3: set_int] :
      ( ( S3 != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_113_set__notEmptyE,axiom,
    ! [S3: set_o] :
      ( ( S3 != bot_bot_set_o )
     => ~ ! [X3: $o] :
            ~ ( member_o @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_114_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_115_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_116_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_117_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_118_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_119_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X222: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X222 ) ) ) ).

% VEBT.exhaust
thf(fact_120_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_121_finite_OemptyI,axiom,
    finite6017078050557962740nteger @ bot_bo3990330152332043303nteger ).

% finite.emptyI
thf(fact_122_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_123_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_124_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_125_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_126_infinite__imp__nonempty,axiom,
    ! [S3: set_Code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S3 )
     => ( S3 != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
thf(fact_127_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_128_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ ( finite_finite_int @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_129_infinite__imp__nonempty,axiom,
    ! [S3: set_o] :
      ( ~ ( finite_finite_o @ S3 )
     => ( S3 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_130_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_131_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_132_mod__h__bot__indep,axiom,
    ! [P2: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_133_zero__reorient,axiom,
    ! [X: uint32] :
      ( ( zero_zero_uint32 = X )
      = ( X = zero_zero_uint32 ) ) ).

% zero_reorient
thf(fact_134_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_135_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_136_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_137_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_138_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_139_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_140_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_141_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_142_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_143_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_144_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_145_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_146_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_147_invar__vebt__buildup,axiom,
    ! [N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% invar_vebt_buildup
thf(fact_148_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).

% buildup_gives_valid
thf(fact_149_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_150_set__vebt__buildup,axiom,
    ! [I: nat] :
      ( ( vEBT_set_vebt @ ( vEBT_vebt_buildup @ I ) )
      = bot_bot_set_nat ) ).

% set_vebt_buildup
thf(fact_151_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_152_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_153_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_154_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A4: set_nat] :
      ( ( ord_less_nat @ Z @ X )
     => ( ( vEBT_VEBT_min_in_set @ A4 @ Z )
       => ( ( finite_finite_nat @ A4 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A4 @ X @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_155_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A4 @ Z )
       => ( ( finite_finite_nat @ B4 )
         => ( ( A4 = B4 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A4 @ X @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_156_finite__psubset__induct,axiom,
    ! [A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A5: set_nat] :
            ( ( finite_finite_nat @ A5 )
           => ( ! [B5: set_nat] :
                  ( ( ord_less_set_nat @ B5 @ A5 )
                 => ( P2 @ B5 ) )
             => ( P2 @ A5 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_157_finite__psubset__induct,axiom,
    ! [A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ! [A5: set_int] :
            ( ( finite_finite_int @ A5 )
           => ( ! [B5: set_int] :
                  ( ( ord_less_set_int @ B5 @ A5 )
                 => ( P2 @ B5 ) )
             => ( P2 @ A5 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_158_finite__psubset__induct,axiom,
    ! [A4: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [A5: set_complex] :
            ( ( finite3207457112153483333omplex @ A5 )
           => ( ! [B5: set_complex] :
                  ( ( ord_less_set_complex @ B5 @ A5 )
                 => ( P2 @ B5 ) )
             => ( P2 @ A5 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_159_finite__psubset__induct,axiom,
    ! [A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [A5: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ A5 )
           => ( ! [B5: set_Code_integer] :
                  ( ( ord_le1307284697595431911nteger @ B5 @ A5 )
                 => ( P2 @ B5 ) )
             => ( P2 @ A5 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_160_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_161_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_162_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_163_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_164_infinite__descent,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P2 @ N2 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
              & ~ ( P2 @ M2 ) ) )
     => ( P2 @ N ) ) ).

% infinite_descent
thf(fact_165_nat__less__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ( P2 @ M2 ) )
         => ( P2 @ N2 ) )
     => ( P2 @ N ) ) ).

% nat_less_induct
thf(fact_166_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_167_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_168_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_169_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_170_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_171_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_172_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_173_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_174_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_175_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_176_zero__induct,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ! [N2: nat] :
            ( ( P2 @ ( suc @ N2 ) )
           => ( P2 @ N2 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_177_diff__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P2 @ X3 @ Y3 )
             => ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P2 @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_178_nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P2 @ N2 )
           => ( P2 @ ( suc @ N2 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_induct
thf(fact_179_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_180_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_181_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_182_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_183_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_184_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_185_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P2 @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P2 @ ( suc @ I2 ) )
               => ( P2 @ I2 ) ) )
         => ( P2 @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_186_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P2 @ I2 @ J2 )
                 => ( ( P2 @ J2 @ K2 )
                   => ( P2 @ I2 @ K2 ) ) ) ) )
         => ( P2 @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_187_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_188_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_189_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_190_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M4: nat] :
            ( ( M
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_191_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P2 @ I3 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_192_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_193_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_194_Ex__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P2 @ I3 ) ) )
      = ( ( P2 @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P2 @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_195_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_196_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_197_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_198_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_199_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_200_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_201_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P2 @ N2 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N2 )
                  & ~ ( P2 @ M2 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_202_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_203_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_204_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_205_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_206_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_207_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_208_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_209_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_210_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_211_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_212_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_213_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_214_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_215_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_216_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_217_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_218_Ex__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_219_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M5: nat] :
            ( N
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_220_All__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_221_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_222_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_223_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_224_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_225_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ X @ Z2 ) )
           => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ).

% succ_member
thf(fact_226_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ Z2 @ X ) )
           => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ).

% pred_member
thf(fact_227_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_228_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_229_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_230_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ X3 ) ) ) ).

% VEBT_internal.insert'.cases
thf(fact_231_forall__finite_I3_J,axiom,
    ! [X: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ X ) ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X ) )
           => ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_232_forall__finite_I2_J,axiom,
    ! [P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ zero_zero_nat ) )
           => ( P2 @ I3 ) ) )
      = ( P2 @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_233_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ X )
           => ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_234_infinite__growing,axiom,
    ! [X6: set_Code_integer] :
      ( ( X6 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X6 )
           => ? [Xa: code_integer] :
                ( ( member_Code_integer @ Xa @ X6 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa ) ) )
       => ~ ( finite6017078050557962740nteger @ X6 ) ) ) ).

% infinite_growing
thf(fact_235_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ X6 )
           => ? [Xa: $o] :
                ( ( member_o @ Xa @ X6 )
                & ( ord_less_o @ X3 @ Xa ) ) )
       => ~ ( finite_finite_o @ X6 ) ) ) ).

% infinite_growing
thf(fact_236_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X6 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X6 )
                & ( ord_less_real @ X3 @ Xa ) ) )
       => ~ ( finite_finite_real @ X6 ) ) ) ).

% infinite_growing
thf(fact_237_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X6 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X6 )
                & ( ord_less_rat @ X3 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X6 ) ) ) ).

% infinite_growing
thf(fact_238_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X6 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X6 )
                & ( ord_less_num @ X3 @ Xa ) ) )
       => ~ ( finite_finite_num @ X6 ) ) ) ).

% infinite_growing
thf(fact_239_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X6 )
                & ( ord_less_nat @ X3 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X6 ) ) ) ).

% infinite_growing
thf(fact_240_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X6 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X6 )
                & ( ord_less_int @ X3 @ Xa ) ) )
       => ~ ( finite_finite_int @ X6 ) ) ) ).

% infinite_growing
thf(fact_241_ex__min__if__finite,axiom,
    ! [S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S3 )
            & ~ ? [Xa: code_integer] :
                  ( ( member_Code_integer @ Xa @ S3 )
                  & ( ord_le6747313008572928689nteger @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_242_ex__min__if__finite,axiom,
    ! [S3: set_o] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ S3 )
            & ~ ? [Xa: $o] :
                  ( ( member_o @ Xa @ S3 )
                  & ( ord_less_o @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_243_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_244_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S3 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S3 )
                  & ( ord_less_rat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_245_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S3 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S3 )
                  & ( ord_less_num @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_246_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_247_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_248_empty__Collect__eq,axiom,
    ! [P2: complex > $o] :
      ( ( bot_bot_set_complex
        = ( collect_complex @ P2 ) )
      = ( ! [X4: complex] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_249_empty__Collect__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P2 ) )
      = ( ! [X4: list_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_250_empty__Collect__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P2 ) )
      = ( ! [X4: set_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_251_empty__Collect__eq,axiom,
    ! [P2: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P2 ) )
      = ( ! [X4: nat] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_252_empty__Collect__eq,axiom,
    ! [P2: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P2 ) )
      = ( ! [X4: int] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_253_empty__Collect__eq,axiom,
    ! [P2: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P2 ) )
      = ( ! [X4: $o] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_254_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs2: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs2 )
          & ! [Y4: nat] :
              ( ( member_nat @ Y4 @ Xs2 )
             => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).

% max_in_set_def
thf(fact_255_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs2: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs2 )
          & ! [Y4: nat] :
              ( ( member_nat @ Y4 @ Xs2 )
             => ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% min_in_set_def
thf(fact_256_empty__iff,axiom,
    ! [C2: vEBT_VEBT] :
      ~ ( member_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ).

% empty_iff
thf(fact_257_empty__iff,axiom,
    ! [C2: real] :
      ~ ( member_real @ C2 @ bot_bot_set_real ) ).

% empty_iff
thf(fact_258_empty__iff,axiom,
    ! [C2: set_nat] :
      ~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_259_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_260_empty__iff,axiom,
    ! [C2: int] :
      ~ ( member_int @ C2 @ bot_bot_set_int ) ).

% empty_iff
thf(fact_261_empty__iff,axiom,
    ! [C2: $o] :
      ~ ( member_o @ C2 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_262_all__not__in__conv,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( ! [X4: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X4 @ A4 ) )
      = ( A4 = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
thf(fact_263_all__not__in__conv,axiom,
    ! [A4: set_real] :
      ( ( ! [X4: real] :
            ~ ( member_real @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_264_all__not__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ( ! [X4: set_nat] :
            ~ ( member_set_nat @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_265_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X4: nat] :
            ~ ( member_nat @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_266_all__not__in__conv,axiom,
    ! [A4: set_int] :
      ( ( ! [X4: int] :
            ~ ( member_int @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_267_all__not__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ! [X4: $o] :
            ~ ( member_o @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_268_Collect__empty__eq,axiom,
    ! [P2: complex > $o] :
      ( ( ( collect_complex @ P2 )
        = bot_bot_set_complex )
      = ( ! [X4: complex] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_269_Collect__empty__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( ( collect_list_nat @ P2 )
        = bot_bot_set_list_nat )
      = ( ! [X4: list_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_270_Collect__empty__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( ( collect_set_nat @ P2 )
        = bot_bot_set_set_nat )
      = ( ! [X4: set_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_271_Collect__empty__eq,axiom,
    ! [P2: nat > $o] :
      ( ( ( collect_nat @ P2 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_272_Collect__empty__eq,axiom,
    ! [P2: int > $o] :
      ( ( ( collect_int @ P2 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_273_Collect__empty__eq,axiom,
    ! [P2: $o > $o] :
      ( ( ( collect_o @ P2 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_274_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_275_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_276_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_277_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_278_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_279_psubsetD,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A4 @ B4 )
     => ( ( member_VEBT_VEBT @ C2 @ A4 )
       => ( member_VEBT_VEBT @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_280_psubsetD,axiom,
    ! [A4: set_real,B4: set_real,C2: real] :
      ( ( ord_less_set_real @ A4 @ B4 )
     => ( ( member_real @ C2 @ A4 )
       => ( member_real @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_281_psubsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ( ( member_int @ C2 @ A4 )
       => ( member_int @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_282_psubsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( ( ord_less_set_set_nat @ A4 @ B4 )
     => ( ( member_set_nat @ C2 @ A4 )
       => ( member_set_nat @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_283_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P2 @ X3 )
            & ! [Y5: nat] :
                ( ( P2 @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_284_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_285_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_286_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_287_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_288_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_289_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ( C2 = D )
         => ( ord_less_eq_set_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_290_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ( C2 = D )
         => ( ord_less_eq_rat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_291_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ( C2 = D )
         => ( ord_less_eq_num @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_292_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( C2 = D )
         => ( ord_less_eq_nat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_293_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ( C2 = D )
         => ( ord_less_eq_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_294_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_295_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_296_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_297_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_298_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_299_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_300_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_301_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_302_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_303_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_304_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_305_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ( ord_less_eq_real @ A @ X3 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_306_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ A @ X3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_307_finite__has__maximal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A @ A4 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ( ord_le3102999989581377725nteger @ A @ X3 )
            & ! [Xa: code_integer] :
                ( ( member_Code_integer @ Xa @ A4 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_308_finite__has__maximal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( member_set_int @ A @ A4 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ( ord_less_eq_set_int @ A @ X3 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_309_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_310_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_311_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_312_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_313_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ( ord_less_eq_real @ X3 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_314_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ X3 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_315_finite__has__minimal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A @ A4 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ( ord_le3102999989581377725nteger @ X3 @ A )
            & ! [Xa: code_integer] :
                ( ( member_Code_integer @ Xa @ A4 )
               => ( ( ord_le3102999989581377725nteger @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_316_finite__has__minimal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( member_set_int @ A @ A4 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ( ord_less_eq_set_int @ X3 @ A )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_317_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_318_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_319_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_320_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_321_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_322_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_323_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_324_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M3: nat] :
          ( M6
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_325_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_326_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_327_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_328_full__nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
             => ( P2 @ M2 ) )
         => ( P2 @ N2 ) )
     => ( P2 @ N ) ) ).

% full_nat_induct
thf(fact_329_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P2 @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P2 @ N2 )
               => ( P2 @ ( suc @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_330_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
           => ( R2 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_331_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_332_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_333_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_334_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_335_exists__leI,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ N )
           => ~ ( P2 @ N4 ) )
       => ( P2 @ N ) )
     => ? [N5: nat] :
          ( ( ord_less_eq_nat @ N5 @ N )
          & ( P2 @ N5 ) ) ) ).

% exists_leI
thf(fact_336_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N6: nat] :
          ( ( ord_less_eq_nat @ M5 @ N6 )
          & ( M5 != N6 ) ) ) ) ).

% nat_less_le
thf(fact_337_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_338_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N6: nat] :
          ( ( ord_less_nat @ M5 @ N6 )
          | ( M5 = N6 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_339_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_340_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_341_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_342_finite__has__minimal,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ! [Xa: code_integer] :
                ( ( member_Code_integer @ Xa @ A4 )
               => ( ( ord_le3102999989581377725nteger @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_343_finite__has__minimal,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A4 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A4 )
               => ( ( ord_less_eq_o @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_344_finite__has__minimal,axiom,
    ! [A4: set_set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_345_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_346_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_347_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_348_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_349_finite__has__maximal,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ! [Xa: code_integer] :
                ( ( member_Code_integer @ Xa @ A4 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_350_finite__has__maximal,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A4 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A4 )
               => ( ( ord_less_eq_o @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_351_finite__has__maximal,axiom,
    ! [A4: set_set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_352_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_353_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_354_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_355_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_356_nat__compl__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P2 @ Nn ) )
           => ( P2 @ ( suc @ N2 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_compl_induct
thf(fact_357_nat__compl__induct_H,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P2 @ Nn ) )
           => ( P2 @ ( suc @ N2 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_compl_induct'
thf(fact_358_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_359_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_360_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_361_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_362_dec__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P2 @ N2 )
                 => ( P2 @ ( suc @ N2 ) ) ) ) )
         => ( P2 @ J ) ) ) ) ).

% dec_induct
thf(fact_363_inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P2 @ ( suc @ N2 ) )
                 => ( P2 @ N2 ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% inc_induct
thf(fact_364_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_365_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_366_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_367_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N6: nat] : ( ord_less_eq_nat @ ( suc @ N6 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_368_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_369_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_370_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_371_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_372_emptyE,axiom,
    ! [A: vEBT_VEBT] :
      ~ ( member_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ).

% emptyE
thf(fact_373_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_374_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_375_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_376_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_377_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_378_equals0D,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( A4 = bot_bo8194388402131092736T_VEBT )
     => ~ ( member_VEBT_VEBT @ A @ A4 ) ) ).

% equals0D
thf(fact_379_equals0D,axiom,
    ! [A4: set_real,A: real] :
      ( ( A4 = bot_bot_set_real )
     => ~ ( member_real @ A @ A4 ) ) ).

% equals0D
thf(fact_380_equals0D,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( A4 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_381_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_382_equals0D,axiom,
    ! [A4: set_int,A: int] :
      ( ( A4 = bot_bot_set_int )
     => ~ ( member_int @ A @ A4 ) ) ).

% equals0D
thf(fact_383_equals0D,axiom,
    ! [A4: set_o,A: $o] :
      ( ( A4 = bot_bot_set_o )
     => ~ ( member_o @ A @ A4 ) ) ).

% equals0D
thf(fact_384_equals0I,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ! [Y3: vEBT_VEBT] :
          ~ ( member_VEBT_VEBT @ Y3 @ A4 )
     => ( A4 = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
thf(fact_385_equals0I,axiom,
    ! [A4: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real @ Y3 @ A4 )
     => ( A4 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_386_equals0I,axiom,
    ! [A4: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_387_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_388_equals0I,axiom,
    ! [A4: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A4 )
     => ( A4 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_389_equals0I,axiom,
    ! [A4: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A4 )
     => ( A4 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_390_ex__in__conv,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( ? [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ A4 ) )
      = ( A4 != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
thf(fact_391_ex__in__conv,axiom,
    ! [A4: set_real] :
      ( ( ? [X4: real] : ( member_real @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_392_ex__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ( ? [X4: set_nat] : ( member_set_nat @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_393_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_394_ex__in__conv,axiom,
    ! [A4: set_int] :
      ( ( ? [X4: int] : ( member_int @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_395_ex__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ? [X4: $o] : ( member_o @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_396_not__psubset__empty,axiom,
    ! [A4: set_nat] :
      ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_397_not__psubset__empty,axiom,
    ! [A4: set_int] :
      ~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_398_not__psubset__empty,axiom,
    ! [A4: set_o] :
      ~ ( ord_less_set_o @ A4 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_399_forall__finite_I1_J,axiom,
    ! [P2: nat > $o,I4: nat] :
      ( ( ord_less_nat @ I4 @ zero_zero_nat )
     => ( P2 @ I4 ) ) ).

% forall_finite(1)
thf(fact_400_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_401_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X4: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X4 )
          | ( vEBT_VEBT_membermima @ T2 @ X4 ) ) ) ) ).

% both_member_options_def
thf(fact_402_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs2: set_nat,X4: nat,Y4: nat] :
          ( ( member_nat @ Y4 @ Xs2 )
          & ( ord_less_nat @ X4 @ Y4 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs2 )
             => ( ( ord_less_nat @ X4 @ Z2 )
               => ( ord_less_eq_nat @ Y4 @ Z2 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_403_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs2: set_nat,X4: nat,Y4: nat] :
          ( ( member_nat @ Y4 @ Xs2 )
          & ( ord_less_nat @ Y4 @ X4 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs2 )
             => ( ( ord_less_nat @ Z2 @ X4 )
               => ( ord_less_eq_nat @ Z2 @ Y4 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_404_order__refl,axiom,
    ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).

% order_refl
thf(fact_405_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_406_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_407_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_408_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_409_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_410_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_411_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_412_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_413_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_414_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_415_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu2: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu2 @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_416_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv2 ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_417_empty__subsetI,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).

% empty_subsetI
thf(fact_418_empty__subsetI,axiom,
    ! [A4: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A4 ) ).

% empty_subsetI
thf(fact_419_empty__subsetI,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).

% empty_subsetI
thf(fact_420_subset__empty,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_421_subset__empty,axiom,
    ! [A4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ bot_bot_set_o )
      = ( A4 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_422_subset__empty,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
      = ( A4 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_423_psubsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( A4 != B4 )
       => ( ord_less_set_int @ A4 @ B4 ) ) ) ).

% psubsetI
thf(fact_424_bot__set__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex @ bot_bot_complex_o ) ) ).

% bot_set_def
thf(fact_425_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_426_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_427_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_428_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_429_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_430_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_431_subset__emptyI,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ~ ( member_VEBT_VEBT @ X3 @ A4 )
     => ( ord_le4337996190870823476T_VEBT @ A4 @ bot_bo8194388402131092736T_VEBT ) ) ).

% subset_emptyI
thf(fact_432_subset__emptyI,axiom,
    ! [A4: set_real] :
      ( ! [X3: real] :
          ~ ( member_real @ X3 @ A4 )
     => ( ord_less_eq_set_real @ A4 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_433_subset__emptyI,axiom,
    ! [A4: set_set_nat] :
      ( ! [X3: set_nat] :
          ~ ( member_set_nat @ X3 @ A4 )
     => ( ord_le6893508408891458716et_nat @ A4 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_434_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A4 )
     => ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_435_subset__emptyI,axiom,
    ! [A4: set_o] :
      ( ! [X3: $o] :
          ~ ( member_o @ X3 @ A4 )
     => ( ord_less_eq_set_o @ A4 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_436_subset__emptyI,axiom,
    ! [A4: set_int] :
      ( ! [X3: int] :
          ~ ( member_int @ X3 @ A4 )
     => ( ord_less_eq_set_int @ A4 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_437_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_438_rev__finite__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( finite3207457112153483333omplex @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_439_rev__finite__subset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( finite6017078050557962740nteger @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_440_rev__finite__subset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( finite_finite_int @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_441_infinite__super,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_442_infinite__super,axiom,
    ! [S3: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S3 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_443_infinite__super,axiom,
    ! [S3: set_Code_integer,T3: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
     => ( ~ ( finite6017078050557962740nteger @ S3 )
       => ~ ( finite6017078050557962740nteger @ T3 ) ) ) ).

% infinite_super
thf(fact_444_infinite__super,axiom,
    ! [S3: set_int,T3: set_int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ~ ( finite_finite_int @ S3 )
       => ~ ( finite_finite_int @ T3 ) ) ) ).

% infinite_super
thf(fact_445_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( finite_finite_nat @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_446_finite__subset,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ B4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( finite3207457112153483333omplex @ A4 ) ) ) ).

% finite_subset
thf(fact_447_finite__subset,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( finite6017078050557962740nteger @ A4 ) ) ) ).

% finite_subset
thf(fact_448_finite__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( finite_finite_int @ B4 )
       => ( finite_finite_int @ A4 ) ) ) ).

% finite_subset
thf(fact_449_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H2 ) ).

% mod_false
thf(fact_450_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_451_subset__psubset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( ord_less_set_int @ B4 @ C3 )
       => ( ord_less_set_int @ A4 @ C3 ) ) ) ).

% subset_psubset_trans
thf(fact_452_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_453_psubset__subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ C3 )
       => ( ord_less_set_int @ A4 @ C3 ) ) ) ).

% psubset_subset_trans
thf(fact_454_psubset__imp__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ( ord_less_eq_set_int @ A4 @ B4 ) ) ).

% psubset_imp_subset
thf(fact_455_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_456_psubsetE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ~ ( ( ord_less_eq_set_int @ A4 @ B4 )
         => ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ).

% psubsetE
thf(fact_457_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_458_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_459_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_460_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_461_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_462_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_463_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_464_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_465_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_466_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_467_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_468_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_469_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_470_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_471_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_472_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_473_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_474_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_475_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_476_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_477_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_478_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_479_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_480_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_481_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_482_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_483_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_484_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_485_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_486_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_487_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_488_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_489_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_490_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_491_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_492_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_493_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_494_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_495_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_496_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_497_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_498_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_499_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_500_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_501_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_502_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_503_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_504_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_505_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_506_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_507_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_508_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_509_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C2: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_510_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_511_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_512_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_513_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C2: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_514_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_515_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C2: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_516_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
    = ( ^ [A7: set_int,B7: set_int] :
          ( ( ord_less_eq_set_int @ A7 @ B7 )
          & ( ord_less_eq_set_int @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_517_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A7 @ B7 )
          & ( ord_less_eq_rat @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_518_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ A7 @ B7 )
          & ( ord_less_eq_num @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_519_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A7 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_520_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ A7 @ B7 )
          & ( ord_less_eq_int @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_521_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_522_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_523_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_524_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_525_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_526_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C2 @ B )
       => ( ord_less_eq_set_int @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_527_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_528_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ B )
       => ( ord_less_eq_num @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_529_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_530_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_531_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_532_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_533_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_534_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_535_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_536_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
    = ( ^ [A7: set_int,B7: set_int] :
          ( ( ord_less_eq_set_int @ B7 @ A7 )
          & ( ord_less_eq_set_int @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_537_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ B7 @ A7 )
          & ( ord_less_eq_rat @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_538_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ B7 @ A7 )
          & ( ord_less_eq_num @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_539_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A7 )
          & ( ord_less_eq_nat @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_540_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ B7 @ A7 )
          & ( ord_less_eq_int @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_541_linorder__wlog,axiom,
    ! [P2: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: rat,B3: rat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_542_linorder__wlog,axiom,
    ! [P2: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: num,B3: num] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_543_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_544_linorder__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_545_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_eq_set_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_546_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X @ Z ) ) ) ).

% order_trans
thf(fact_547_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X @ Z ) ) ) ).

% order_trans
thf(fact_548_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_549_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_550_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% order.trans
thf(fact_551_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_552_order_Otrans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% order.trans
thf(fact_553_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_554_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% order.trans
thf(fact_555_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_556_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_557_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_558_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_559_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_560_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_561_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_562_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_563_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_564_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_565_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_566_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_567_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_568_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_569_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_570_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y4 )
          & ( ord_less_eq_set_int @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_571_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_572_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_573_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_574_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_575_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z ) )
       => ( ( ( ord_less_eq_rat @ X @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_576_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z ) )
       => ( ( ( ord_less_eq_num @ X @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X ) )
             => ~ ( ( ord_less_eq_num @ Z @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_577_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_578_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_579_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B ) )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_580_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_581_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_582_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_583_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_584_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_585_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_586_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_587_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_588_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_589_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_590_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_591_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_592_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_593_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_594_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_595_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_596_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_597_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_598_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_599_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
      | ( X = Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_600_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
      | ( X = Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_601_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_602_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_603_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P2: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_604_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P2: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_605_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P2: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_606_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_607_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P2: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_608_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_609_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_610_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_611_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_612_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_613_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_614_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C2: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_615_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C2: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_616_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_617_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C2: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_618_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C2: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_619_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_620_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_621_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_622_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_623_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C2: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_624_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C2: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_625_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C2: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_626_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C2: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_627_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C2: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_628_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C2: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_629_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_630_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_631_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_632_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C2: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_633_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_634_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_635_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_636_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_637_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_638_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_639_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C2: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_640_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C2: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_641_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_642_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C2: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_643_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C2: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_644_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_645_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_646_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_647_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_648_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C2: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_649_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C2: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_650_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C2: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_651_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C2: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_652_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C2: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_653_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_654_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_655_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_656_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_657_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_658_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_659_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_660_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_661_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_662_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_663_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_664_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_665_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_666_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_667_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_668_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_669_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      = ( ( ord_less_rat @ X @ Y )
        | ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_670_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      = ( ( ord_less_num @ X @ Y )
        | ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_671_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_672_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_673_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_674_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_675_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_676_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_677_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_678_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_679_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_680_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_681_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_682_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_683_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_684_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_685_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_686_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_687_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_688_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_689_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_690_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_691_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_692_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_693_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C2 @ B )
       => ( ord_less_real @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_694_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_695_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_696_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_697_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_698_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_699_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_700_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ( ord_less_num @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_701_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_702_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_703_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_704_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_705_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_706_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_707_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_708_linorder__less__wlog,axiom,
    ! [P2: real > real > $o,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: real] : ( P2 @ A3 @ A3 )
       => ( ! [A3: real,B3: real] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_709_linorder__less__wlog,axiom,
    ! [P2: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: rat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: rat,B3: rat] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_710_linorder__less__wlog,axiom,
    ! [P2: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: num] : ( P2 @ A3 @ A3 )
       => ( ! [A3: num,B3: num] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_711_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_712_linorder__less__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P2 @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_713_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X7: nat] : ( P4 @ X7 ) )
    = ( ^ [P5: nat > $o] :
        ? [N6: nat] :
          ( ( P5 @ N6 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N6 )
             => ~ ( P5 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_714_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_715_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_716_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_717_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_718_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_719_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_720_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_721_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_722_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_723_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_724_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_725_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_726_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_727_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_728_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_729_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_730_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_731_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_732_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_733_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_734_less__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P2 @ Y5 ) )
         => ( P2 @ X3 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_735_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_736_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_737_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_738_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_739_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_740_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_741_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_742_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_743_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_744_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_745_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_746_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_747_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_748_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_749_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_750_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_751_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_752_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_753_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_754_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_755_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z3: real] :
          ( ( ord_less_real @ X @ Z3 )
          & ( ord_less_real @ Z3 @ Y ) ) ) ).

% dense
thf(fact_756_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z3: rat] :
          ( ( ord_less_rat @ X @ Z3 )
          & ( ord_less_rat @ Z3 @ Y ) ) ) ).

% dense
thf(fact_757_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).

% gt_ex
thf(fact_758_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_759_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_760_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_761_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_762_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_763_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_764_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_765_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_766_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_767_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_768_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_769_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_770_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_771_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_772_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_773_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_774_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_775_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_776_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C2: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_777_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C2: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_778_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C2: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_779_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C2: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_780_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C2: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_781_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_782_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_783_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C2: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_784_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C2: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_785_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C2: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_786_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_787_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_788_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_789_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C2: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_790_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B: num,C2: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_791_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_792_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C2: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_793_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C2: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_794_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C2: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_795_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C2: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_796_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_797_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_798_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_799_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_800_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > real,C2: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_801_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_802_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_803_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_804_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_805_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_806_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C2: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_807_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C2: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_808_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C2: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_809_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C2: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_810_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C2: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_811_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_812_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_813_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_814_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C2: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_815_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_816_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_817_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_818_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_819_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_820_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_821_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_822_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_823_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_824_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_825_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_826_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_827_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_828_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_829_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_830_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( ( ord_less_eq_num @ A @ B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_831_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_832_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_833_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_834_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_835_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( A != B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_836_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( A != B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_837_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_838_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_839_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_840_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_841_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_842_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_843_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_844_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_845_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_846_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_847_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_848_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_849_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_850_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_851_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_852_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_853_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_854_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_855_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y4: real] :
          ( ( ord_less_eq_real @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_856_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_857_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_858_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_859_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_860_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_861_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y4: real] :
          ( ( ord_less_real @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_862_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_set_int @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_863_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_rat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_864_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_num @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_865_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_nat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_866_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_int @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_867_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_868_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ord_less_eq_set_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_869_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_870_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ord_less_eq_num @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_871_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_872_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_873_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_874_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_875_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_876_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ord_less_eq_num @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_877_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_878_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_879_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A7: real] :
          ( ( ord_less_eq_real @ B7 @ A7 )
          & ~ ( ord_less_eq_real @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_880_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B7: set_int,A7: set_int] :
          ( ( ord_less_eq_set_int @ B7 @ A7 )
          & ~ ( ord_less_eq_set_int @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_881_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_less_eq_rat @ B7 @ A7 )
          & ~ ( ord_less_eq_rat @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_882_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_less_eq_num @ B7 @ A7 )
          & ~ ( ord_less_eq_num @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_883_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A7 )
          & ~ ( ord_less_eq_nat @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_884_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_less_eq_int @ B7 @ A7 )
          & ~ ( ord_less_eq_int @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_885_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C2 @ B )
       => ( ord_less_real @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_886_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C2 @ B )
       => ( ord_less_set_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_887_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_888_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_889_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_890_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_891_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C2 @ B )
       => ( ord_less_real @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_892_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_set_int @ C2 @ B )
       => ( ord_less_set_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_893_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_894_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_895_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_896_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_897_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A7: real] :
          ( ( ord_less_eq_real @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_898_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B7: set_int,A7: set_int] :
          ( ( ord_less_eq_set_int @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_899_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_less_eq_rat @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_900_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_less_eq_num @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_901_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_902_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_less_eq_int @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_903_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B7: real,A7: real] :
          ( ( ord_less_real @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_904_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B7: set_int,A7: set_int] :
          ( ( ord_less_set_int @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_905_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_less_rat @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_906_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_less_num @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_907_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_less_nat @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_908_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_less_int @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_909_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_910_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X @ W )
           => ( ( ord_less_rat @ W @ Y )
             => ( ord_less_eq_rat @ W @ Z ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_911_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ X )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z @ W )
           => ( ( ord_less_real @ W @ X )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_912_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ X )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z @ W )
           => ( ( ord_less_rat @ W @ X )
             => ( ord_less_eq_rat @ Y @ W ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_913_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A7: real,B7: real] :
          ( ( ord_less_eq_real @ A7 @ B7 )
          & ~ ( ord_less_eq_real @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_914_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A7: set_int,B7: set_int] :
          ( ( ord_less_eq_set_int @ A7 @ B7 )
          & ~ ( ord_less_eq_set_int @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_915_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A7 @ B7 )
          & ~ ( ord_less_eq_rat @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_916_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ A7 @ B7 )
          & ~ ( ord_less_eq_num @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_917_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A7 @ B7 )
          & ~ ( ord_less_eq_nat @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_918_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ A7 @ B7 )
          & ~ ( ord_less_eq_int @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_919_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_920_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ord_less_set_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_921_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_922_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_923_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_924_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_925_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_926_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_set_int @ B @ C2 )
       => ( ord_less_set_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_927_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_928_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_929_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_930_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_931_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A7: real,B7: real] :
          ( ( ord_less_eq_real @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_932_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A7: set_int,B7: set_int] :
          ( ( ord_less_eq_set_int @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_933_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_934_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_935_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_936_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_937_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A7: real,B7: real] :
          ( ( ord_less_real @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_938_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A7: set_int,B7: set_int] :
          ( ( ord_less_set_int @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_939_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_rat @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_940_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_num @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_941_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_nat @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_942_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_int @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_943_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_eq_real @ Y @ X )
     => ( ord_less_real @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_944_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_rat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_945_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_eq_num @ Y @ X )
     => ( ord_less_num @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_946_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_947_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_948_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y4: real] :
          ( ( ord_less_eq_real @ X4 @ Y4 )
          & ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_949_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y4 )
          & ~ ( ord_less_eq_set_int @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_950_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ~ ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_951_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ~ ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_952_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_953_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_954_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ X3 @ Y )
         => ( ord_less_eq_real @ X3 @ Z ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_le
thf(fact_955_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y )
         => ( ord_less_eq_rat @ X3 @ Z ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_le
thf(fact_956_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ Z @ X3 )
         => ( ord_less_eq_real @ Y @ X3 ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_ge
thf(fact_957_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z @ X3 )
         => ( ord_less_eq_rat @ Y @ X3 ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_ge
thf(fact_958_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_959_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ~ ( ord_less_set_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_960_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_961_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_962_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_963_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_964_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_965_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_966_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_967_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_968_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_969_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_970_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_971_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ~ ( ord_less_set_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_972_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_973_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_974_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_975_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_976_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_977_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_978_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_979_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_980_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_981_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_982_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ~ ( ord_less_set_int @ X @ Y ) ) ).

% leD
thf(fact_983_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_984_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_985_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_986_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_987_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_988_bot_Oextremum__uniqueI,axiom,
    ! [A: assn] :
      ( ( ord_less_eq_assn @ A @ bot_bot_assn )
     => ( A = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
thf(fact_989_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_990_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_991_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_992_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_993_bot_Oextremum__unique,axiom,
    ! [A: assn] :
      ( ( ord_less_eq_assn @ A @ bot_bot_assn )
      = ( A = bot_bot_assn ) ) ).

% bot.extremum_unique
thf(fact_994_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_995_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_996_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_997_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_998_bot_Oextremum,axiom,
    ! [A: assn] : ( ord_less_eq_assn @ bot_bot_assn @ A ) ).

% bot.extremum
thf(fact_999_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_1000_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1001_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1002_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1003_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
      = ( ord_less_assn @ bot_bot_assn @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1004_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1005_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1006_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1007_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1008_bot_Oextremum__strict,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ bot_bot_assn ) ).

% bot.extremum_strict
thf(fact_1009_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1010_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_1011_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1012_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1013_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A7: $o,B7: $o] :
            ( T
            = ( vEBT_Leaf @ A7 @ B7 ) ) ) ) ).

% deg1Leaf
thf(fact_1014_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A3: $o,B3: $o] :
          ( T
          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).

% deg_1_Leaf
thf(fact_1015_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg_1_Leafy
thf(fact_1016_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N7: set_nat] :
        ? [M5: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N7 )
         => ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1017_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N6: nat] :
            ( ( ord_less_eq_nat @ M5 @ N6 )
            & ( member_nat @ N6 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1018_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N7: set_nat] :
        ? [M5: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N7 )
         => ( ord_less_nat @ X4 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1019_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N6: nat] :
            ( ( ord_less_nat @ M5 @ N6 )
            & ( member_nat @ N6 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1020_bounded__nat__set__is__finite,axiom,
    ! [N8: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N8 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N8 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1021_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ K @ M3 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M3 @ N9 )
              & ( member_nat @ N9 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_1022_VEBT__internal_Oheight_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).

% VEBT_internal.height.cases
thf(fact_1023_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X5: complex] :
              ( ( member_complex @ X5 @ S3 )
              & ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1024_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ~ ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ S3 )
              & ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic2659822949269061924r_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1025_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ S3 )
              & ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1026_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X5: int] :
              ( ( member_int @ X5 @ S3 )
              & ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1027_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X5: $o] :
              ( ( member_o @ X5 @ S3 )
              & ( ord_less_real @ ( F @ X5 ) @ ( F @ ( lattic8697145971487455083o_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1028_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X5: complex] :
              ( ( member_complex @ X5 @ S3 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1029_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ~ ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ S3 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic5439806495466278992er_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1030_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ S3 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1031_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X5: int] :
              ( ( member_int @ X5 @ S3 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1032_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X5: $o] :
              ( ( member_o @ X5 @ S3 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1033_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( member_nat @ X3 @ B4 ) )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_1034_subsetI,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( member_VEBT_VEBT @ X3 @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ A4 @ B4 ) ) ).

% subsetI
thf(fact_1035_subsetI,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( member_real @ X3 @ B4 ) )
     => ( ord_less_eq_set_real @ A4 @ B4 ) ) ).

% subsetI
thf(fact_1036_subsetI,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A4 )
         => ( member_set_nat @ X3 @ B4 ) )
     => ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_1037_subsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( member_int @ X3 @ B4 ) )
     => ( ord_less_eq_set_int @ A4 @ B4 ) ) ).

% subsetI
thf(fact_1038_subset__antisym,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_1039_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1040_subset__Collect__conv,axiom,
    ! [S3: set_complex,P2: complex > $o] :
      ( ( ord_le211207098394363844omplex @ S3 @ ( collect_complex @ P2 ) )
      = ( ! [X4: complex] :
            ( ( member_complex @ X4 @ S3 )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1041_subset__Collect__conv,axiom,
    ! [S3: set_list_nat,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ S3 @ ( collect_list_nat @ P2 ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ S3 )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1042_subset__Collect__conv,axiom,
    ! [S3: set_set_nat,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ S3 @ ( collect_set_nat @ P2 ) )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ S3 )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1043_subset__Collect__conv,axiom,
    ! [S3: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ S3 @ ( collect_nat @ P2 ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ S3 )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1044_subset__Collect__conv,axiom,
    ! [S3: set_int,P2: int > $o] :
      ( ( ord_less_eq_set_int @ S3 @ ( collect_int @ P2 ) )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ S3 )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1045_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ X @ A4 )
       => ( member_nat @ X @ B4 ) ) ) ).

% in_mono
thf(fact_1046_in__mono,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( member_VEBT_VEBT @ X @ B4 ) ) ) ).

% in_mono
thf(fact_1047_in__mono,axiom,
    ! [A4: set_real,B4: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A4 @ B4 )
     => ( ( member_real @ X @ A4 )
       => ( member_real @ X @ B4 ) ) ) ).

% in_mono
thf(fact_1048_in__mono,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( member_set_nat @ X @ B4 ) ) ) ).

% in_mono
thf(fact_1049_in__mono,axiom,
    ! [A4: set_int,B4: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( member_int @ X @ A4 )
       => ( member_int @ X @ B4 ) ) ) ).

% in_mono
thf(fact_1050_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_1051_subsetD,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
     => ( ( member_VEBT_VEBT @ C2 @ A4 )
       => ( member_VEBT_VEBT @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_1052_subsetD,axiom,
    ! [A4: set_real,B4: set_real,C2: real] :
      ( ( ord_less_eq_set_real @ A4 @ B4 )
     => ( ( member_real @ C2 @ A4 )
       => ( member_real @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_1053_subsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
     => ( ( member_set_nat @ C2 @ A4 )
       => ( member_set_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_1054_subsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( member_int @ C2 @ A4 )
       => ( member_int @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_1055_equalityE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => ~ ( ( ord_less_eq_set_int @ A4 @ B4 )
         => ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ).

% equalityE
thf(fact_1056_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ( member_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1057_subset__eq,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
        ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ A6 )
         => ( member_VEBT_VEBT @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1058_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X4: real] :
          ( ( member_real @ X4 @ A6 )
         => ( member_real @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1059_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A6 )
         => ( member_set_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1060_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X4: int] :
          ( ( member_int @ X4 @ A6 )
         => ( member_int @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1061_equalityD1,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => ( ord_less_eq_set_int @ A4 @ B4 ) ) ).

% equalityD1
thf(fact_1062_Set_OequalityD2,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => ( ord_less_eq_set_int @ B4 @ A4 ) ) ).

% Set.equalityD2
thf(fact_1063_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_1064_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_1065_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_1066_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat @ T2 @ A6 )
         => ( member_set_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1067_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_1068_subset__refl,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).

% subset_refl
thf(fact_1069_Collect__mono,axiom,
    ! [P2: complex > $o,Q2: complex > $o] :
      ( ! [X3: complex] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P2 ) @ ( collect_complex @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1070_Collect__mono,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1071_Collect__mono,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1072_Collect__mono,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1073_Collect__mono,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1074_subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ C3 )
       => ( ord_less_eq_set_int @ A4 @ C3 ) ) ) ).

% subset_trans
thf(fact_1075_set__eq__subset,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = 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_1076_Collect__mono__iff,axiom,
    ! [P2: complex > $o,Q2: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P2 ) @ ( collect_complex @ Q2 ) )
      = ( ! [X4: complex] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1077_Collect__mono__iff,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) )
      = ( ! [X4: list_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1078_Collect__mono__iff,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) )
      = ( ! [X4: set_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1079_Collect__mono__iff,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
      = ( ! [X4: nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1080_Collect__mono__iff,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) )
      = ( ! [X4: int] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1081_one__reorient,axiom,
    ! [X: uint32] :
      ( ( one_one_uint32 = X )
      = ( X = one_one_uint32 ) ) ).

% one_reorient
thf(fact_1082_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_1083_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_1084_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_1085_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_1086_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_1087_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1088_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X )
      = ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_1089_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_1090_nat__induct__non__zero,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P2 @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P2 @ N2 )
               => ( P2 @ ( suc @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1091_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X: nat,M7: nat] :
      ( ( P2 @ X )
     => ( ! [X3: nat] :
            ( ( P2 @ X3 )
           => ( ord_less_eq_nat @ X3 @ M7 ) )
       => ~ ! [M3: nat] :
              ( ( P2 @ M3 )
             => ~ ! [X5: nat] :
                    ( ( P2 @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1092_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_1093_finite__transitivity__chain,axiom,
    ! [A4: set_VEBT_VEBT,R2: vEBT_VEBT > vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: vEBT_VEBT,Y3: vEBT_VEBT,Z3: vEBT_VEBT] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ A4 )
               => ? [Y5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1094_finite__transitivity__chain,axiom,
    ! [A4: set_real,R2: real > real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: real,Y3: real,Z3: real] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ A4 )
               => ? [Y5: real] :
                    ( ( member_real @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1095_finite__transitivity__chain,axiom,
    ! [A4: set_set_nat,R2: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ! [X3: set_nat] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: set_nat,Y3: set_nat,Z3: set_nat] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: set_nat] :
                ( ( member_set_nat @ X3 @ A4 )
               => ? [Y5: set_nat] :
                    ( ( member_set_nat @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1096_finite__transitivity__chain,axiom,
    ! [A4: set_complex,R2: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: complex,Y3: complex,Z3: complex] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A4 )
               => ? [Y5: complex] :
                    ( ( member_complex @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1097_finite__transitivity__chain,axiom,
    ! [A4: set_Code_integer,R2: code_integer > code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: code_integer,Y3: code_integer,Z3: code_integer] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ A4 )
               => ? [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bo3990330152332043303nteger ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1098_finite__transitivity__chain,axiom,
    ! [A4: set_nat,R2: nat > nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A4 )
               => ? [Y5: nat] :
                    ( ( member_nat @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1099_finite__transitivity__chain,axiom,
    ! [A4: set_int,R2: int > int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: int,Y3: int,Z3: int] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ A4 )
               => ? [Y5: int] :
                    ( ( member_int @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1100_finite__transitivity__chain,axiom,
    ! [A4: set_o,R2: $o > $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ! [X3: $o] :
            ~ ( R2 @ X3 @ X3 )
       => ( ! [X3: $o,Y3: $o,Z3: $o] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ A4 )
               => ? [Y5: $o] :
                    ( ( member_o @ Y5 @ A4 )
                    & ( R2 @ X3 @ Y5 ) ) )
           => ( A4 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1101_arg__min__least,axiom,
    ! [S3: set_VEBT_VEBT,Y: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( S3 != bot_bo8194388402131092736T_VEBT )
       => ( ( member_VEBT_VEBT @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic6139528642216935859BT_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1102_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1103_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1104_arg__min__least,axiom,
    ! [S3: set_Code_integer,Y: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( member_Code_integer @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic5439806495466278992er_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1105_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1106_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1107_arg__min__least,axiom,
    ! [S3: set_o,Y: $o,F: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ( ( member_o @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1108_arg__min__least,axiom,
    ! [S3: set_VEBT_VEBT,Y: vEBT_VEBT,F: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( S3 != bot_bo8194388402131092736T_VEBT )
       => ( ( member_VEBT_VEBT @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic3331990488459210229BT_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1109_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1110_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1111_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ one_one_nat ) ) ) ).

% minNull_delete_time_bound'
thf(fact_1112_vebt__member__code_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member_code(1)
thf(fact_1113_vebt__delete__code_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $false @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $false ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_delete_code(1)
thf(fact_1114_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_1115_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_1116_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1117_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_1118_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_1119_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_1120_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1121_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1122_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_1123_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1124_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1125_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1126_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_1127_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_1128_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_1129_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_1130_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_1131_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_1132_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_1133_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_1134_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1135_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1136_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1137_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1138_bot__empty__eq,axiom,
    ( bot_bot_VEBT_VEBT_o
    = ( ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% bot_empty_eq
thf(fact_1139_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X4: real] : ( member_real @ X4 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_1140_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X4: set_nat] : ( member_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1141_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1142_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X4: int] : ( member_int @ X4 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_1143_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X4: $o] : ( member_o @ X4 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_1144_Collect__empty__eq__bot,axiom,
    ! [P2: complex > $o] :
      ( ( ( collect_complex @ P2 )
        = bot_bot_set_complex )
      = ( P2 = bot_bot_complex_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1145_Collect__empty__eq__bot,axiom,
    ! [P2: list_nat > $o] :
      ( ( ( collect_list_nat @ P2 )
        = bot_bot_set_list_nat )
      = ( P2 = bot_bot_list_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1146_Collect__empty__eq__bot,axiom,
    ! [P2: set_nat > $o] :
      ( ( ( collect_set_nat @ P2 )
        = bot_bot_set_set_nat )
      = ( P2 = bot_bot_set_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1147_Collect__empty__eq__bot,axiom,
    ! [P2: nat > $o] :
      ( ( ( collect_nat @ P2 )
        = bot_bot_set_nat )
      = ( P2 = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1148_Collect__empty__eq__bot,axiom,
    ! [P2: int > $o] :
      ( ( ( collect_int @ P2 )
        = bot_bot_set_int )
      = ( P2 = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1149_Collect__empty__eq__bot,axiom,
    ! [P2: $o > $o] :
      ( ( ( collect_o @ P2 )
        = bot_bot_set_o )
      = ( P2 = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1150_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_1151_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1152_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1153_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1154_subrelI,axiom,
    ! [R: set_Pr1773385645901665561uint32,S: set_Pr1773385645901665561uint32] :
      ( ! [X3: uint32,Y3: uint32] :
          ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ Y3 ) @ R )
         => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ Y3 ) @ S ) )
     => ( ord_le6429528607962791097uint32 @ R @ S ) ) ).

% subrelI
thf(fact_1155_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% subrelI
thf(fact_1156_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R @ S ) ) ).

% subrelI
thf(fact_1157_subrelI,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
          ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R )
         => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le5966269811547037844et_nat @ R @ S ) ) ).

% subrelI
thf(fact_1158_subrelI,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
          ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R )
         => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le4763372923235995152et_nat @ R @ S ) ) ).

% subrelI
thf(fact_1159_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_1160_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1161_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1162_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1163_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1164_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1165_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_1166_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1167_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1168_zero__neq__one,axiom,
    zero_zero_uint32 != one_one_uint32 ).

% zero_neq_one
thf(fact_1169_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1170_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1171_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1172_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1173_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_1174_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_1175_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1176_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1177_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_1178_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_1179_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1180_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1181_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_1182_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_1183_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,N: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_1184_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P2 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K2 ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_1185_exists__least__lemma,axiom,
    ! [P2: nat > $o] :
      ( ~ ( P2 @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P2 @ X_12 )
       => ? [N2: nat] :
            ( ~ ( P2 @ N2 )
            & ( P2 @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1186_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,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_1187_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_1188_mod__h__bot__iff_I4_J,axiom,
    ! [Q3: array_VEBT_VEBTi,Y: list_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ~ ( rep_assn @ ( snga_assn_VEBT_VEBTi @ Q3 @ Y ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(4)
thf(fact_1189_complete__interval,axiom,
    ! [A: real,B: real,P2: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: real] :
              ( ( ord_less_eq_real @ A @ C )
              & ( ord_less_eq_real @ C @ B )
              & ! [X5: real] :
                  ( ( ( ord_less_eq_real @ A @ X5 )
                    & ( ord_less_real @ X5 @ C ) )
                 => ( P2 @ X5 ) )
              & ! [D3: real] :
                  ( ! [X3: real] :
                      ( ( ( ord_less_eq_real @ A @ X3 )
                        & ( ord_less_real @ X3 @ D3 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_real @ D3 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_1190_complete__interval,axiom,
    ! [A: nat,B: nat,P2: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: nat] :
              ( ( ord_less_eq_nat @ A @ C )
              & ( ord_less_eq_nat @ C @ B )
              & ! [X5: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X5 )
                    & ( ord_less_nat @ X5 @ C ) )
                 => ( P2 @ X5 ) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D3 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_nat @ D3 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_1191_complete__interval,axiom,
    ! [A: int,B: int,P2: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: int] :
              ( ( ord_less_eq_int @ A @ C )
              & ( ord_less_eq_int @ C @ B )
              & ! [X5: int] :
                  ( ( ( ord_less_eq_int @ A @ X5 )
                    & ( ord_less_int @ X5 @ C ) )
                 => ( P2 @ X5 ) )
              & ! [D3: int] :
                  ( ! [X3: int] :
                      ( ( ( ord_less_eq_int @ A @ X3 )
                        & ( ord_less_int @ X3 @ D3 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_int @ D3 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_1192_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ( ~ ( ord_less_eq_real @ B2 @ A2 ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1193_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ~ ( ord_less_eq_rat @ B2 @ A2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1194_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
      = ( ord_less_num @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1195_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1196_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1197_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1198_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1199_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1200_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1201_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1202_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_1203_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_1204_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_1205_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_1206_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1207_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1208_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1209_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1210_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1211_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( ( ord_less_real @ A @ B3 )
      | ( ord_less_real @ B3 @ A ) ) ).

% ex_gt_or_lt
thf(fact_1212_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X5 ) ) ).

% minf(8)
thf(fact_1213_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).

% minf(8)
thf(fact_1214_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X5 ) ) ).

% minf(8)
thf(fact_1215_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_1216_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X5 ) ) ).

% minf(8)
thf(fact_1217_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z3 )
     => ( ord_less_eq_real @ X5 @ T ) ) ).

% minf(6)
thf(fact_1218_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z3 )
     => ( ord_less_eq_rat @ X5 @ T ) ) ).

% minf(6)
thf(fact_1219_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z3 )
     => ( ord_less_eq_num @ X5 @ T ) ) ).

% minf(6)
thf(fact_1220_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_1221_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z3 )
     => ( ord_less_eq_int @ X5 @ T ) ) ).

% minf(6)
thf(fact_1222_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z3 @ X5 )
     => ( ord_less_eq_real @ T @ X5 ) ) ).

% pinf(8)
thf(fact_1223_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z3 @ X5 )
     => ( ord_less_eq_rat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_1224_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z3 @ X5 )
     => ( ord_less_eq_num @ T @ X5 ) ) ).

% pinf(8)
thf(fact_1225_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_1226_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z3 @ X5 )
     => ( ord_less_eq_int @ T @ X5 ) ) ).

% pinf(8)
thf(fact_1227_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z3 @ X5 )
     => ~ ( ord_less_eq_real @ X5 @ T ) ) ).

% pinf(6)
thf(fact_1228_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z3 @ X5 )
     => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_1229_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z3 @ X5 )
     => ~ ( ord_less_eq_num @ X5 @ T ) ) ).

% pinf(6)
thf(fact_1230_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_1231_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z3 @ X5 )
     => ~ ( ord_less_eq_int @ X5 @ T ) ) ).

% pinf(6)
thf(fact_1232_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E: real] :
            ( ( ord_less_real @ zero_zero_real @ E )
            & ( ord_less_real @ E @ D1 )
            & ( ord_less_real @ E @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_1233_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E )
            & ( ord_less_rat @ E @ D1 )
            & ( ord_less_rat @ E @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_1234_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu4269007558841261821uint32 @ zero_zero_uint32 )
    = one_one_uint32 ) ).

% dbl_inc_simps(2)
thf(fact_1235_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_1236_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_1237_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_1238_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_1239_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_1240_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,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_1241_frac__eq,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = X )
      = ( ( ord_less_eq_real @ zero_zero_real @ X )
        & ( ord_less_real @ X @ one_one_real ) ) ) ).

% frac_eq
thf(fact_1242_frac__eq,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = X )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_1243_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1244_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_1245_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_1246_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_1247_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_1248_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_1249_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1250_diff__zero,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% diff_zero
thf(fact_1251_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1252_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1253_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1254_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1255_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1256_diff__0__right,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% diff_0_right
thf(fact_1257_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1258_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1259_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1260_diff__self,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ A )
      = zero_zero_uint32 ) ).

% diff_self
thf(fact_1261_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1262_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1263_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1264_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1265_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1266_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1267_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1268_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_1269_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_1270_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_1271_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_1272_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_1273_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_1274_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_1275_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_1276_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_1277_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_1278_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_1279_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_1280_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1281_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_uint32 @ one_one_uint32 @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% diff_numeral_special(9)
thf(fact_1282_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1283_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1284_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1285_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1286_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1287_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1288_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1289_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N @ M ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_1290_diff__eq__diff__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32,D: uint32] :
      ( ( ( minus_minus_uint32 @ A @ B )
        = ( minus_minus_uint32 @ C2 @ D ) )
     => ( ( A = B )
        = ( C2 = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1291_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C2 @ D ) )
     => ( ( A = B )
        = ( C2 = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1292_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D ) )
     => ( ( A = B )
        = ( C2 = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1293_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: uint32,C2: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ ( minus_minus_uint32 @ A @ C2 ) @ B )
      = ( minus_minus_uint32 @ ( minus_minus_uint32 @ A @ B ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1294_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1295_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1296_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1297_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_1298_le__some__optE,axiom,
    ! [M: set_int,X: option_set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ M ) @ X )
     => ~ ! [M8: set_int] :
            ( ( X
              = ( some_set_int @ M8 ) )
           => ~ ( ord_less_eq_set_int @ M @ M8 ) ) ) ).

% le_some_optE
thf(fact_1299_le__some__optE,axiom,
    ! [M: rat,X: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M ) @ X )
     => ~ ! [M8: rat] :
            ( ( X
              = ( some_rat @ M8 ) )
           => ~ ( ord_less_eq_rat @ M @ M8 ) ) ) ).

% le_some_optE
thf(fact_1300_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X )
     => ~ ! [M8: num] :
            ( ( X
              = ( some_num @ M8 ) )
           => ~ ( ord_less_eq_num @ M @ M8 ) ) ) ).

% le_some_optE
thf(fact_1301_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X )
     => ~ ! [M8: nat] :
            ( ( X
              = ( some_nat @ M8 ) )
           => ~ ( ord_less_eq_nat @ M @ M8 ) ) ) ).

% le_some_optE
thf(fact_1302_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X )
     => ~ ! [M8: int] :
            ( ( X
              = ( some_int @ M8 ) )
           => ~ ( ord_less_eq_int @ M @ M8 ) ) ) ).

% le_some_optE
thf(fact_1303_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [A7: rat,B7: rat] :
          ( ( minus_minus_rat @ A7 @ B7 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1304_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: uint32,Z4: uint32] : ( Y6 = Z4 ) )
    = ( ^ [A7: uint32,B7: uint32] :
          ( ( minus_minus_uint32 @ A7 @ B7 )
          = zero_zero_uint32 ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1305_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
    = ( ^ [A7: real,B7: real] :
          ( ( minus_minus_real @ A7 @ B7 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1306_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [A7: int,B7: int] :
          ( ( minus_minus_int @ A7 @ B7 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1307_diff__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C2 )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1308_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C2 )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1309_diff__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C2 )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1310_diff__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_1311_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_1312_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_1313_diff__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1314_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1315_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1316_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C2 @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C2 @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1317_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C2 @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C2 @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1318_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C2 @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1319_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_1320_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_1321_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_1322_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1323_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1324_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1325_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C2 @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C2 @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1326_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C2 @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C2 @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1327_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C2 @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1328_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C2 )
       => ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1329_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C2 )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1330_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C2 )
       => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1331_zero__induct__lemma,axiom,
    ! [P2: nat > $o,K: nat,I: nat] :
      ( ( P2 @ K )
     => ( ! [N2: nat] :
            ( ( P2 @ ( suc @ N2 ) )
           => ( P2 @ N2 ) )
       => ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1332_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1333_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1334_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1335_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1336_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1337_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1338_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1339_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1340_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1341_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1342_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1343_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,X: 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 ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_1344_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A7: real,B7: real] : ( ord_less_eq_real @ ( minus_minus_real @ A7 @ B7 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_1345_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A7: rat,B7: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A7 @ B7 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1346_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A7: int,B7: int] : ( ord_less_eq_int @ ( minus_minus_int @ A7 @ B7 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1347_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A7: real,B7: real] : ( ord_less_real @ ( minus_minus_real @ A7 @ B7 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_1348_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A7: rat,B7: rat] : ( ord_less_rat @ ( minus_minus_rat @ A7 @ B7 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_1349_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A7: int,B7: int] : ( ord_less_int @ ( minus_minus_int @ A7 @ B7 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1350_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( ( suc @ N )
        = M )
     => ( N
        = ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_1351_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1352_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1353_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1354_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1355_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1356_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1357_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1358_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,X: 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 ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_1359_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_1360_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_1361_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_1362_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_1363_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_1364_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1365_pinf_I1_J,axiom,
    ! [P2: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1366_pinf_I1_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1367_pinf_I1_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1368_pinf_I1_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1369_pinf_I1_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1370_pinf_I2_J,axiom,
    ! [P2: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1371_pinf_I2_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1372_pinf_I2_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1373_pinf_I2_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1374_pinf_I2_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1375_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_1376_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_1377_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_1378_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_1379_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_1380_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_1381_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_1382_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_1383_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_1384_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z3 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_1385_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z3 @ X5 )
     => ~ ( ord_less_real @ X5 @ T ) ) ).

% pinf(5)
thf(fact_1386_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z3 @ X5 )
     => ~ ( ord_less_rat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_1387_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z3 @ X5 )
     => ~ ( ord_less_num @ X5 @ T ) ) ).

% pinf(5)
thf(fact_1388_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ~ ( ord_less_nat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_1389_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z3 @ X5 )
     => ~ ( ord_less_int @ X5 @ T ) ) ).

% pinf(5)
thf(fact_1390_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z3 @ X5 )
     => ( ord_less_real @ T @ X5 ) ) ).

% pinf(7)
thf(fact_1391_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z3 @ X5 )
     => ( ord_less_rat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_1392_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z3 @ X5 )
     => ( ord_less_num @ T @ X5 ) ) ).

% pinf(7)
thf(fact_1393_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ( ord_less_nat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_1394_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z3 @ X5 )
     => ( ord_less_int @ T @ X5 ) ) ).

% pinf(7)
thf(fact_1395_minf_I1_J,axiom,
    ! [P2: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1396_minf_I1_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1397_minf_I1_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1398_minf_I1_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1399_minf_I1_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                & ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1400_minf_I2_J,axiom,
    ! [P2: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1401_minf_I2_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1402_minf_I2_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1403_minf_I2_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1404_minf_I2_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( ( P2 @ X5 )
                | ( Q2 @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q4 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1405_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_1406_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_1407_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_1408_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_1409_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_1410_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_1411_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_1412_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_1413_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_1414_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z3 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_1415_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z3 )
     => ( ord_less_real @ X5 @ T ) ) ).

% minf(5)
thf(fact_1416_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z3 )
     => ( ord_less_rat @ X5 @ T ) ) ).

% minf(5)
thf(fact_1417_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z3 )
     => ( ord_less_num @ X5 @ T ) ) ).

% minf(5)
thf(fact_1418_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ( ord_less_nat @ X5 @ T ) ) ).

% minf(5)
thf(fact_1419_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z3 )
     => ( ord_less_int @ X5 @ T ) ) ).

% minf(5)
thf(fact_1420_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z3 )
     => ~ ( ord_less_real @ T @ X5 ) ) ).

% minf(7)
thf(fact_1421_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z3 )
     => ~ ( ord_less_rat @ T @ X5 ) ) ).

% minf(7)
thf(fact_1422_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z3 )
     => ~ ( ord_less_num @ T @ X5 ) ) ).

% minf(7)
thf(fact_1423_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ~ ( ord_less_nat @ T @ X5 ) ) ).

% minf(7)
thf(fact_1424_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z3 )
     => ~ ( ord_less_int @ T @ X5 ) ) ).

% minf(7)
thf(fact_1425_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_1426_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_1427_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_1428_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_1429_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_1430_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1431_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1432_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N @ M ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_1433_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_1434_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_1435_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,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_1436_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,N: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_1437_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_1438_less__option__Some,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_option_real @ ( some_real @ X ) @ ( some_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% less_option_Some
thf(fact_1439_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_rat @ X @ Y ) ) ).

% less_option_Some
thf(fact_1440_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_option_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_num @ X @ Y ) ) ).

% less_option_Some
thf(fact_1441_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_option_Some
thf(fact_1442_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_int @ X @ Y ) ) ).

% less_option_Some
thf(fact_1443_less__eq__option__Some,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ X ) @ ( some_set_int @ Y ) )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1444_less__eq__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1445_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1446_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1447_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1448_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_1449_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( minus_minus_assn @ X @ Y )
        = bot_bot_assn )
      = ( ord_less_eq_assn @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1450_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1451_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1452_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1453_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K3 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1454_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1455_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ X3 ) )
           => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_1456_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,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,B3: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,N2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Uu ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( 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] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_1457_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N ) ) )
      = ( ( ord_less_nat @ I @ M )
        & ( ord_less_nat @ I @ N ) ) ) ).

% diff_diff_less
thf(fact_1458_Diff__empty,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ bot_bot_set_int )
      = A4 ) ).

% Diff_empty
thf(fact_1459_Diff__empty,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ bot_bot_set_o )
      = A4 ) ).

% Diff_empty
thf(fact_1460_Diff__empty,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Diff_empty
thf(fact_1461_empty__Diff,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A4 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_1462_empty__Diff,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A4 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_1463_empty__Diff,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_1464_Diff__cancel,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ A4 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_1465_Diff__cancel,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ A4 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_1466_Diff__cancel,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ A4 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_1467_finite__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_1468_finite__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_1469_finite__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_1470_finite__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_1471_finite__Diff2,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B4 ) )
        = ( finite_finite_int @ A4 ) ) ) ).

% finite_Diff2
thf(fact_1472_finite__Diff2,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
        = ( finite3207457112153483333omplex @ A4 ) ) ) ).

% finite_Diff2
thf(fact_1473_finite__Diff2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A4 @ B4 ) )
        = ( finite6017078050557962740nteger @ A4 ) ) ) ).

% finite_Diff2
thf(fact_1474_finite__Diff2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
        = ( finite_finite_nat @ A4 ) ) ) ).

% finite_Diff2
thf(fact_1475_Diff__eq__empty__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( minus_minus_set_o @ A4 @ B4 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1476_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1477_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( minus_minus_set_int @ A4 @ B4 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1478_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ).

% lesseq_shift
thf(fact_1479_less__eq__option__None__code,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None_code
thf(fact_1480_less__option__None,axiom,
    ! [X: option_nat] :
      ~ ( ord_less_option_nat @ X @ none_nat ) ).

% less_option_None
thf(fact_1481_not__Some__eq2,axiom,
    ! [V2: option3972171592325465343uint32] :
      ( ( ! [X4: uint32,Y4: uint32] :
            ( V2
           != ( some_P6695178580830101022uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) ) ) )
      = ( V2 = none_P7441528354948028570uint32 ) ) ).

% not_Some_eq2
thf(fact_1482_not__Some__eq2,axiom,
    ! [V2: option4624381673175914239nt_int] :
      ( ( ! [X4: int,Y4: int] :
            ( V2
           != ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) ) ) )
      = ( V2 = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
thf(fact_1483_not__Some__eq2,axiom,
    ! [V2: option5190343406534369742et_nat] :
      ( ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] :
            ( V2
           != ( some_P750831030444334937et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) ) ) )
      = ( V2 = none_P4972525538344268765et_nat ) ) ).

% not_Some_eq2
thf(fact_1484_not__Some__eq2,axiom,
    ! [V2: option2860828798490689354et_nat] :
      ( ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] :
            ( V2
           != ( some_P1630309045189364437et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) ) ) )
      = ( V2 = none_P199884684680593241et_nat ) ) ).

% not_Some_eq2
thf(fact_1485_not__Some__eq2,axiom,
    ! [V2: option4927543243414619207at_nat] :
      ( ( ! [X4: nat,Y4: nat] :
            ( V2
           != ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) )
      = ( V2 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
thf(fact_1486_less__eq__option__Some__None,axiom,
    ! [X: nat] :
      ~ ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ none_nat ) ).

% less_eq_option_Some_None
thf(fact_1487_less__option__None__Some__code,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some_code
thf(fact_1488_less__eq__option__None,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None
thf(fact_1489_less__eq__option__None__is__None,axiom,
    ! [X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ X @ none_nat )
     => ( X = none_nat ) ) ).

% less_eq_option_None_is_None
thf(fact_1490_less__option__None__Some,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some
thf(fact_1491_less__option__None__is__Some,axiom,
    ! [X: option_nat] :
      ( ( ord_less_option_nat @ none_nat @ X )
     => ? [Z3: nat] :
          ( X
          = ( some_nat @ Z3 ) ) ) ).

% less_option_None_is_Some
thf(fact_1492_Diff__infinite__finite,axiom,
    ! [T3: set_int,S3: set_int] :
      ( ( finite_finite_int @ T3 )
     => ( ~ ( finite_finite_int @ S3 )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1493_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S3: set_complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1494_Diff__infinite__finite,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ~ ( finite6017078050557962740nteger @ S3 )
       => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1495_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1496_Diff__mono,axiom,
    ! [A4: set_nat,C3: set_nat,D4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C3 )
     => ( ( ord_less_eq_set_nat @ D4 @ B4 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ C3 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_1497_Diff__mono,axiom,
    ! [A4: set_int,C3: set_int,D4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ C3 )
     => ( ( ord_less_eq_set_int @ D4 @ B4 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B4 ) @ ( minus_minus_set_int @ C3 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_1498_Diff__subset,axiom,
    ! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ A4 ) ).

% Diff_subset
thf(fact_1499_Diff__subset,axiom,
    ! [A4: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B4 ) @ A4 ) ).

% Diff_subset
thf(fact_1500_double__diff,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C3 )
       => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C3 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_1501_double__diff,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ C3 )
       => ( ( minus_minus_set_int @ B4 @ ( minus_minus_set_int @ C3 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_1502_psubset__imp__ex__mem,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A4 @ B4 )
     => ? [B3: vEBT_VEBT] : ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1503_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( ord_less_set_real @ A4 @ B4 )
     => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1504_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1505_psubset__imp__ex__mem,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ( ord_less_set_set_nat @ A4 @ B4 )
     => ? [B3: set_nat] : ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1506_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1507_subset__minus__empty,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ( minus_minus_set_o @ A4 @ B4 )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_1508_subset__minus__empty,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( minus_minus_set_nat @ A4 @ B4 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_1509_subset__minus__empty,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( minus_minus_set_int @ A4 @ B4 )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_1510_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu2 )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_1511_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_1512_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X ) ).

% vebt_member.simps(2)
thf(fact_1513_vebt__member__code_I2_J,axiom,
    ! [T: nat,R: list_VEBT_VEBT,E2: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ T @ R @ E2 ) @ X ) ).

% vebt_member_code(2)
thf(fact_1514_bijective__def,axiom,
    ( biject2923219584331114343uint32
    = ( ^ [R3: set_Pr1773385645901665561uint32] :
          ( ! [X4: uint32,Y4: uint32,Z2: uint32] :
              ( ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) @ R3 )
                & ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Z2 ) @ R3 ) )
             => ( Y4 = Z2 ) )
          & ! [X4: uint32,Y4: uint32,Z2: uint32] :
              ( ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Z2 ) @ R3 )
                & ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y4 @ Z2 ) @ R3 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_1515_bijective__def,axiom,
    ( bijective_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( ! [X4: nat,Y4: nat,Z2: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R3 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Z2 ) @ R3 ) )
             => ( Y4 = Z2 ) )
          & ! [X4: nat,Y4: nat,Z2: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Z2 ) @ R3 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z2 ) @ R3 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_1516_bijective__def,axiom,
    ( bijective_int_int
    = ( ^ [R3: set_Pr958786334691620121nt_int] :
          ( ! [X4: int,Y4: int,Z2: int] :
              ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R3 )
                & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Z2 ) @ R3 ) )
             => ( Y4 = Z2 ) )
          & ! [X4: int,Y4: int,Z2: int] :
              ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Z2 ) @ R3 )
                & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ Z2 ) @ R3 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_1517_bijective__def,axiom,
    ( biject2615096655818420098et_nat
    = ( ^ [R3: set_Pr3286484037609594932et_nat] :
          ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R3 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Z2 ) @ R3 ) )
             => ( Y4 = Z2 ) )
          & ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat > $o,Z2: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Z2 ) @ R3 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y4 @ Z2 ) @ R3 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_1518_bijective__def,axiom,
    ( biject1468766312547416318et_nat
    = ( ^ [R3: set_Pr8536935166611901872et_nat] :
          ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat,Z2: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R3 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Z2 ) @ R3 ) )
             => ( Y4 = Z2 ) )
          & ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat > $o,Z2: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Z2 ) @ R3 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y4 @ Z2 ) @ R3 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_1519_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_1520_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_1521_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,Uu2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu2 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_1522_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,Uu2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu2 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_1523_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => Y )
         => ( ( ? [Uu: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
             => Y )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_1524_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,Va: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) )
         => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf ) )
             => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_1525_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B3: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) )
     => ( ! [Uv: $o,Uw: $o,N2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) )
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) @ Va3 ) )
         => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Ve ) )
           => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_1526_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) @ X3 ) )
         => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_1527_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) )
       => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ X3 ) )
         => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_1528_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_1529_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_1530_List_Ofinite__set,axiom,
    ! [Xs: list_real] : ( finite_finite_real @ ( set_real2 @ Xs ) ) ).

% List.finite_set
thf(fact_1531_List_Ofinite__set,axiom,
    ! [Xs: list_o] : ( finite_finite_o @ ( set_o2 @ Xs ) ) ).

% List.finite_set
thf(fact_1532_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1533_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_1534_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_1535_List_Ofinite__set,axiom,
    ! [Xs: list_Code_integer] : ( finite6017078050557962740nteger @ ( set_Code_integer2 @ Xs ) ) ).

% List.finite_set
thf(fact_1536_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_1537_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_1538_insertCI,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B4 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertCI
thf(fact_1539_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B4 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_1540_insertCI,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ~ ( member_VEBT_VEBT @ A @ B4 )
       => ( A = B ) )
     => ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ B4 ) ) ) ).

% insertCI
thf(fact_1541_insertCI,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B4 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_1542_insertCI,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( ( ~ ( member_int @ A @ B4 )
       => ( A = B ) )
     => ( member_int @ A @ ( insert_int @ B @ B4 ) ) ) ).

% insertCI
thf(fact_1543_insertCI,axiom,
    ! [A: set_nat,B4: set_set_nat,B: set_nat] :
      ( ( ~ ( member_set_nat @ A @ B4 )
       => ( A = B ) )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_1544_insert__iff,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
      = ( ( A = B )
        | ( member_o @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_1545_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_1546_insert__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ A4 ) )
      = ( ( A = B )
        | ( member_VEBT_VEBT @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_1547_insert__iff,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A4 ) )
      = ( ( A = B )
        | ( member_real @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_1548_insert__iff,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A4 ) )
      = ( ( A = B )
        | ( member_int @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_1549_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member_set_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_1550_insert__absorb2,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A4 ) )
      = ( insert_nat @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_1551_insert__absorb2,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ X @ A4 ) )
      = ( insert_VEBT_VEBT @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_1552_insert__absorb2,axiom,
    ! [X: int,A4: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A4 ) )
      = ( insert_int @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_1553_insert__absorb2,axiom,
    ! [X: $o,A4: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A4 ) )
      = ( insert_o @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_1554_DiffI,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ A4 )
     => ( ~ ( member_VEBT_VEBT @ C2 @ B4 )
       => ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1555_DiffI,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C2 @ A4 )
     => ( ~ ( member_real @ C2 @ B4 )
       => ( member_real @ C2 @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1556_DiffI,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C2 @ A4 )
     => ( ~ ( member_int @ C2 @ B4 )
       => ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1557_DiffI,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ C2 @ A4 )
     => ( ~ ( member_set_nat @ C2 @ B4 )
       => ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1558_DiffI,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ A4 )
     => ( ~ ( member_nat @ C2 @ B4 )
       => ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1559_Diff__iff,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) )
      = ( ( member_VEBT_VEBT @ C2 @ A4 )
        & ~ ( member_VEBT_VEBT @ C2 @ B4 ) ) ) ).

% Diff_iff
thf(fact_1560_Diff__iff,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A4 @ B4 ) )
      = ( ( member_real @ C2 @ A4 )
        & ~ ( member_real @ C2 @ B4 ) ) ) ).

% Diff_iff
thf(fact_1561_Diff__iff,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B4 ) )
      = ( ( member_int @ C2 @ A4 )
        & ~ ( member_int @ C2 @ B4 ) ) ) ).

% Diff_iff
thf(fact_1562_Diff__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) )
      = ( ( member_set_nat @ C2 @ A4 )
        & ~ ( member_set_nat @ C2 @ B4 ) ) ) ).

% Diff_iff
thf(fact_1563_Diff__iff,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C2 @ A4 )
        & ~ ( member_nat @ C2 @ B4 ) ) ) ).

% Diff_iff
thf(fact_1564_Diff__idemp,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ B4 )
      = ( minus_minus_set_nat @ A4 @ B4 ) ) ).

% Diff_idemp
thf(fact_1565_set__vebt__delete,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_vebt_delete
thf(fact_1566_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_1567_singletonI,axiom,
    ! [A: vEBT_VEBT] : ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singletonI
thf(fact_1568_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_1569_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_1570_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_1571_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_1572_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_1573_finite__insert,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ ( insert_VEBT_VEBT @ A @ A4 ) )
      = ( finite5795047828879050333T_VEBT @ A4 ) ) ).

% finite_insert
thf(fact_1574_finite__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( ( finite_finite_o @ ( insert_o @ A @ A4 ) )
      = ( finite_finite_o @ A4 ) ) ).

% finite_insert
thf(fact_1575_finite__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A @ A4 ) )
      = ( finite_finite_nat @ A4 ) ) ).

% finite_insert
thf(fact_1576_finite__insert,axiom,
    ! [A: int,A4: set_int] :
      ( ( finite_finite_int @ ( insert_int @ A @ A4 ) )
      = ( finite_finite_int @ A4 ) ) ).

% finite_insert
thf(fact_1577_finite__insert,axiom,
    ! [A: complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A4 ) )
      = ( finite3207457112153483333omplex @ A4 ) ) ).

% finite_insert
thf(fact_1578_finite__insert,axiom,
    ! [A: code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( insert_Code_integer @ A @ A4 ) )
      = ( finite6017078050557962740nteger @ A4 ) ) ).

% finite_insert
thf(fact_1579_insert__subset,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X @ A4 ) @ B4 )
      = ( ( member_o @ X @ B4 )
        & ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1580_insert__subset,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
      = ( ( member_nat @ X @ B4 )
        & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1581_insert__subset,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) @ B4 )
      = ( ( member_VEBT_VEBT @ X @ B4 )
        & ( ord_le4337996190870823476T_VEBT @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1582_insert__subset,axiom,
    ! [X: real,A4: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A4 ) @ B4 )
      = ( ( member_real @ X @ B4 )
        & ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1583_insert__subset,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A4 ) @ B4 )
      = ( ( member_set_nat @ X @ B4 )
        & ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1584_insert__subset,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A4 ) @ B4 )
      = ( ( member_int @ X @ B4 )
        & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1585_Diff__insert0,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ( minus_minus_set_o @ A4 @ ( insert_o @ X @ B4 ) )
        = ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1586_Diff__insert0,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A4 )
     => ( ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ B4 ) )
        = ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1587_Diff__insert0,axiom,
    ! [X: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X @ A4 )
     => ( ( minus_minus_set_real @ A4 @ ( insert_real @ X @ B4 ) )
        = ( minus_minus_set_real @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1588_Diff__insert0,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ( minus_minus_set_int @ A4 @ ( insert_int @ X @ B4 ) )
        = ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1589_Diff__insert0,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ B4 ) )
        = ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1590_Diff__insert0,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1591_insert__Diff1,axiom,
    ! [X: $o,B4: set_o,A4: set_o] :
      ( ( member_o @ X @ B4 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B4 )
        = ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1592_insert__Diff1,axiom,
    ! [X: vEBT_VEBT,B4: set_VEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ B4 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) @ B4 )
        = ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1593_insert__Diff1,axiom,
    ! [X: real,B4: set_real,A4: set_real] :
      ( ( member_real @ X @ B4 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A4 ) @ B4 )
        = ( minus_minus_set_real @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1594_insert__Diff1,axiom,
    ! [X: int,B4: set_int,A4: set_int] :
      ( ( member_int @ X @ B4 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ B4 )
        = ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1595_insert__Diff1,axiom,
    ! [X: set_nat,B4: set_set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ X @ B4 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ B4 )
        = ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1596_insert__Diff1,axiom,
    ! [X: nat,B4: set_nat,A4: set_nat] :
      ( ( member_nat @ X @ B4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1597_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A @ A4 )
        = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A4 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1598_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A4 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1599_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A4: set_o,B: $o] :
      ( ( ( insert_o @ A @ A4 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1600_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A4: set_int,B: int] :
      ( ( ( insert_int @ A @ A4 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1601_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT )
        = ( insert_VEBT_VEBT @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A4 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1602_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1603_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A4: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1604_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A4: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1605_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( insert_VEBT_VEBT @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1606_insert__Diff__single,axiom,
    ! [A: int,A4: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1607_insert__Diff__single,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1608_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1609_finite__Diff__insert,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ B4 ) ) )
      = ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1610_finite__Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( finite_finite_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) ) )
      = ( finite_finite_o @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1611_finite__Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B4 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1612_finite__Diff__insert,axiom,
    ! [A4: set_complex,A: complex,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ B4 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1613_finite__Diff__insert,axiom,
    ! [A4: set_Code_integer,A: code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ A @ B4 ) ) )
      = ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1614_finite__Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1615_DiffE,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) )
     => ~ ( ( member_VEBT_VEBT @ C2 @ A4 )
         => ( member_VEBT_VEBT @ C2 @ B4 ) ) ) ).

% DiffE
thf(fact_1616_DiffE,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ~ ( ( member_real @ C2 @ A4 )
         => ( member_real @ C2 @ B4 ) ) ) ).

% DiffE
thf(fact_1617_DiffE,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ~ ( ( member_int @ C2 @ A4 )
         => ( member_int @ C2 @ B4 ) ) ) ).

% DiffE
thf(fact_1618_DiffE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) )
     => ~ ( ( member_set_nat @ C2 @ A4 )
         => ( member_set_nat @ C2 @ B4 ) ) ) ).

% DiffE
thf(fact_1619_DiffE,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ C2 @ A4 )
         => ( member_nat @ C2 @ B4 ) ) ) ).

% DiffE
thf(fact_1620_DiffD1,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) )
     => ( member_VEBT_VEBT @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_1621_DiffD1,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ( member_real @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_1622_DiffD1,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ( member_int @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_1623_DiffD1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) )
     => ( member_set_nat @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_1624_DiffD1,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ( member_nat @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_1625_DiffD2,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) )
     => ~ ( member_VEBT_VEBT @ C2 @ B4 ) ) ).

% DiffD2
thf(fact_1626_DiffD2,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ~ ( member_real @ C2 @ B4 ) ) ).

% DiffD2
thf(fact_1627_DiffD2,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ~ ( member_int @ C2 @ B4 ) ) ).

% DiffD2
thf(fact_1628_DiffD2,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) )
     => ~ ( member_set_nat @ C2 @ B4 ) ) ).

% DiffD2
thf(fact_1629_DiffD2,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( member_nat @ C2 @ B4 ) ) ).

% DiffD2
thf(fact_1630_insert__Diff__if,axiom,
    ! [X: $o,B4: set_o,A4: set_o] :
      ( ( ( member_o @ X @ B4 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B4 )
          = ( minus_minus_set_o @ A4 @ B4 ) ) )
      & ( ~ ( member_o @ X @ B4 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B4 )
          = ( insert_o @ X @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1631_insert__Diff__if,axiom,
    ! [X: vEBT_VEBT,B4: set_VEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X @ B4 )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) @ B4 )
          = ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) ) )
      & ( ~ ( member_VEBT_VEBT @ X @ B4 )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) @ B4 )
          = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1632_insert__Diff__if,axiom,
    ! [X: real,B4: set_real,A4: set_real] :
      ( ( ( member_real @ X @ B4 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A4 ) @ B4 )
          = ( minus_minus_set_real @ A4 @ B4 ) ) )
      & ( ~ ( member_real @ X @ B4 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A4 ) @ B4 )
          = ( insert_real @ X @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1633_insert__Diff__if,axiom,
    ! [X: int,B4: set_int,A4: set_int] :
      ( ( ( member_int @ X @ B4 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ B4 )
          = ( minus_minus_set_int @ A4 @ B4 ) ) )
      & ( ~ ( member_int @ X @ B4 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ B4 )
          = ( insert_int @ X @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1634_insert__Diff__if,axiom,
    ! [X: set_nat,B4: set_set_nat,A4: set_set_nat] :
      ( ( ( member_set_nat @ X @ B4 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ B4 )
          = ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) )
      & ( ~ ( member_set_nat @ X @ B4 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ B4 )
          = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1635_insert__Diff__if,axiom,
    ! [X: nat,B4: set_nat,A4: set_nat] :
      ( ( ( member_nat @ X @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
          = ( minus_minus_set_nat @ A4 @ B4 ) ) )
      & ( ~ ( member_nat @ X @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1636_insertE,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A4 ) ) ) ).

% insertE
thf(fact_1637_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_1638_insertE,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ A4 ) )
     => ( ( A != B )
       => ( member_VEBT_VEBT @ A @ A4 ) ) ) ).

% insertE
thf(fact_1639_insertE,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A4 ) )
     => ( ( A != B )
       => ( member_real @ A @ A4 ) ) ) ).

% insertE
thf(fact_1640_insertE,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A4 ) )
     => ( ( A != B )
       => ( member_int @ A @ A4 ) ) ) ).

% insertE
thf(fact_1641_insertE,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member_set_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_1642_insertI1,axiom,
    ! [A: $o,B4: set_o] : ( member_o @ A @ ( insert_o @ A @ B4 ) ) ).

% insertI1
thf(fact_1643_insertI1,axiom,
    ! [A: nat,B4: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_1644_insertI1,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ A @ B4 ) ) ).

% insertI1
thf(fact_1645_insertI1,axiom,
    ! [A: real,B4: set_real] : ( member_real @ A @ ( insert_real @ A @ B4 ) ) ).

% insertI1
thf(fact_1646_insertI1,axiom,
    ! [A: int,B4: set_int] : ( member_int @ A @ ( insert_int @ A @ B4 ) ) ).

% insertI1
thf(fact_1647_insertI1,axiom,
    ! [A: set_nat,B4: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_1648_insertI2,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( member_o @ A @ B4 )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1649_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( member_nat @ A @ B4 )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1650_insertI2,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ B4 )
     => ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1651_insertI2,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( member_real @ A @ B4 )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1652_insertI2,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( ( member_int @ A @ B4 )
     => ( member_int @ A @ ( insert_int @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1653_insertI2,axiom,
    ! [A: set_nat,B4: set_set_nat,B: set_nat] :
      ( ( member_set_nat @ A @ B4 )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1654_Set_Oset__insert,axiom,
    ! [X: $o,A4: set_o] :
      ( ( member_o @ X @ A4 )
     => ~ ! [B8: set_o] :
            ( ( A4
              = ( insert_o @ X @ B8 ) )
           => ( member_o @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_1655_Set_Oset__insert,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( member_nat @ X @ A4 )
     => ~ ! [B8: set_nat] :
            ( ( A4
              = ( insert_nat @ X @ B8 ) )
           => ( member_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_1656_Set_Oset__insert,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ~ ! [B8: set_VEBT_VEBT] :
            ( ( A4
              = ( insert_VEBT_VEBT @ X @ B8 ) )
           => ( member_VEBT_VEBT @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_1657_Set_Oset__insert,axiom,
    ! [X: real,A4: set_real] :
      ( ( member_real @ X @ A4 )
     => ~ ! [B8: set_real] :
            ( ( A4
              = ( insert_real @ X @ B8 ) )
           => ( member_real @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_1658_Set_Oset__insert,axiom,
    ! [X: int,A4: set_int] :
      ( ( member_int @ X @ A4 )
     => ~ ! [B8: set_int] :
            ( ( A4
              = ( insert_int @ X @ B8 ) )
           => ( member_int @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_1659_Set_Oset__insert,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ X @ A4 )
     => ~ ! [B8: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ X @ B8 ) )
           => ( member_set_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_1660_insert__ident,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ~ ( member_o @ X @ B4 )
       => ( ( ( insert_o @ X @ A4 )
            = ( insert_o @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1661_insert__ident,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ~ ( member_nat @ X @ B4 )
       => ( ( ( insert_nat @ X @ A4 )
            = ( insert_nat @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1662_insert__ident,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A4 )
     => ( ~ ( member_VEBT_VEBT @ X @ B4 )
       => ( ( ( insert_VEBT_VEBT @ X @ A4 )
            = ( insert_VEBT_VEBT @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1663_insert__ident,axiom,
    ! [X: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X @ A4 )
     => ( ~ ( member_real @ X @ B4 )
       => ( ( ( insert_real @ X @ A4 )
            = ( insert_real @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1664_insert__ident,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ~ ( member_int @ X @ B4 )
       => ( ( ( insert_int @ X @ A4 )
            = ( insert_int @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1665_insert__ident,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ~ ( member_set_nat @ X @ B4 )
       => ( ( ( insert_set_nat @ X @ A4 )
            = ( insert_set_nat @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1666_insert__absorb,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1667_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1668_insert__absorb,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A4 )
     => ( ( insert_VEBT_VEBT @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1669_insert__absorb,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ( insert_real @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1670_insert__absorb,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( insert_int @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1671_insert__absorb,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ( ( insert_set_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1672_insert__eq__iff,axiom,
    ! [A: $o,A4: set_o,B: $o,B4: set_o] :
      ( ~ ( member_o @ A @ A4 )
     => ( ~ ( member_o @ B @ B4 )
       => ( ( ( insert_o @ A @ A4 )
            = ( insert_o @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A = ~ B )
             => ? [C4: set_o] :
                  ( ( A4
                    = ( insert_o @ B @ C4 ) )
                  & ~ ( member_o @ B @ C4 )
                  & ( B4
                    = ( insert_o @ A @ C4 ) )
                  & ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1673_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ A4 )
     => ( ~ ( member_nat @ B @ B4 )
       => ( ( ( insert_nat @ A @ A4 )
            = ( insert_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_nat] :
                  ( ( A4
                    = ( insert_nat @ B @ C4 ) )
                  & ~ ( member_nat @ B @ C4 )
                  & ( B4
                    = ( insert_nat @ A @ C4 ) )
                  & ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1674_insert__eq__iff,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ A @ A4 )
     => ( ~ ( member_VEBT_VEBT @ B @ B4 )
       => ( ( ( insert_VEBT_VEBT @ A @ A4 )
            = ( insert_VEBT_VEBT @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_VEBT_VEBT] :
                  ( ( A4
                    = ( insert_VEBT_VEBT @ B @ C4 ) )
                  & ~ ( member_VEBT_VEBT @ B @ C4 )
                  & ( B4
                    = ( insert_VEBT_VEBT @ A @ C4 ) )
                  & ~ ( member_VEBT_VEBT @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1675_insert__eq__iff,axiom,
    ! [A: real,A4: set_real,B: real,B4: set_real] :
      ( ~ ( member_real @ A @ A4 )
     => ( ~ ( member_real @ B @ B4 )
       => ( ( ( insert_real @ A @ A4 )
            = ( insert_real @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_real] :
                  ( ( A4
                    = ( insert_real @ B @ C4 ) )
                  & ~ ( member_real @ B @ C4 )
                  & ( B4
                    = ( insert_real @ A @ C4 ) )
                  & ~ ( member_real @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1676_insert__eq__iff,axiom,
    ! [A: int,A4: set_int,B: int,B4: set_int] :
      ( ~ ( member_int @ A @ A4 )
     => ( ~ ( member_int @ B @ B4 )
       => ( ( ( insert_int @ A @ A4 )
            = ( insert_int @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_int] :
                  ( ( A4
                    = ( insert_int @ B @ C4 ) )
                  & ~ ( member_int @ B @ C4 )
                  & ( B4
                    = ( insert_int @ A @ C4 ) )
                  & ~ ( member_int @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1677_insert__eq__iff,axiom,
    ! [A: set_nat,A4: set_set_nat,B: set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A4 )
     => ( ~ ( member_set_nat @ B @ B4 )
       => ( ( ( insert_set_nat @ A @ A4 )
            = ( insert_set_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_set_nat] :
                  ( ( A4
                    = ( insert_set_nat @ B @ C4 ) )
                  & ~ ( member_set_nat @ B @ C4 )
                  & ( B4
                    = ( insert_set_nat @ A @ C4 ) )
                  & ~ ( member_set_nat @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1678_insert__commute,axiom,
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ Y @ A4 ) )
      = ( insert_nat @ Y @ ( insert_nat @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_1679_insert__commute,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ Y @ A4 ) )
      = ( insert_VEBT_VEBT @ Y @ ( insert_VEBT_VEBT @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_1680_insert__commute,axiom,
    ! [X: int,Y: int,A4: set_int] :
      ( ( insert_int @ X @ ( insert_int @ Y @ A4 ) )
      = ( insert_int @ Y @ ( insert_int @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_1681_insert__commute,axiom,
    ! [X: $o,Y: $o,A4: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A4 ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_1682_mk__disjoint__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ? [B8: set_o] :
          ( ( A4
            = ( insert_o @ A @ B8 ) )
          & ~ ( member_o @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_1683_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ? [B8: set_nat] :
          ( ( A4
            = ( insert_nat @ A @ B8 ) )
          & ~ ( member_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_1684_mk__disjoint__insert,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A4 )
     => ? [B8: set_VEBT_VEBT] :
          ( ( A4
            = ( insert_VEBT_VEBT @ A @ B8 ) )
          & ~ ( member_VEBT_VEBT @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_1685_mk__disjoint__insert,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ? [B8: set_real] :
          ( ( A4
            = ( insert_real @ A @ B8 ) )
          & ~ ( member_real @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_1686_mk__disjoint__insert,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ? [B8: set_int] :
          ( ( A4
            = ( insert_int @ A @ B8 ) )
          & ~ ( member_int @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_1687_mk__disjoint__insert,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ? [B8: set_set_nat] :
          ( ( A4
            = ( insert_set_nat @ A @ B8 ) )
          & ~ ( member_set_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_1688_singleton__inject,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT )
        = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_1689_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_1690_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_1691_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_1692_insert__not__empty,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A @ A4 )
     != bot_bo8194388402131092736T_VEBT ) ).

% insert_not_empty
thf(fact_1693_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ A4 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_1694_insert__not__empty,axiom,
    ! [A: int,A4: set_int] :
      ( ( insert_int @ A @ A4 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_1695_insert__not__empty,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ A4 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_1696_doubleton__eq__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,C2: vEBT_VEBT,D: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ C2 @ ( insert_VEBT_VEBT @ D @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1697_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C2 @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1698_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C2 @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1699_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C2: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C2 @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1700_singleton__iff,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1701_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1702_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1703_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1704_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1705_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1706_singletonD,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1707_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1708_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1709_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1710_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1711_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1712_insert__minus__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( X != Y )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1713_insert__minus__eq,axiom,
    ! [X: int,Y: int,A4: set_int] :
      ( ( X != Y )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ ( insert_int @ Y @ bot_bot_set_int ) )
        = ( insert_int @ X @ ( minus_minus_set_int @ A4 @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1714_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A4: set_o] :
      ( ( X != Y )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ ( insert_o @ Y @ bot_bot_set_o ) )
        = ( insert_o @ X @ ( minus_minus_set_o @ A4 @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1715_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
        = ( insert_nat @ X @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1716_set__minus__singleton__eq,axiom,
    ! [X: vEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ X6 )
     => ( ( minus_5127226145743854075T_VEBT @ X6 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_1717_set__minus__singleton__eq,axiom,
    ! [X: real,X6: set_real] :
      ( ~ ( member_real @ X @ X6 )
     => ( ( minus_minus_set_real @ X6 @ ( insert_real @ X @ bot_bot_set_real ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_1718_set__minus__singleton__eq,axiom,
    ! [X: set_nat,X6: set_set_nat] :
      ( ~ ( member_set_nat @ X @ X6 )
     => ( ( minus_2163939370556025621et_nat @ X6 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_1719_set__minus__singleton__eq,axiom,
    ! [X: int,X6: set_int] :
      ( ~ ( member_int @ X @ X6 )
     => ( ( minus_minus_set_int @ X6 @ ( insert_int @ X @ bot_bot_set_int ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_1720_set__minus__singleton__eq,axiom,
    ! [X: $o,X6: set_o] :
      ( ~ ( member_o @ X @ X6 )
     => ( ( minus_minus_set_o @ X6 @ ( insert_o @ X @ bot_bot_set_o ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_1721_set__minus__singleton__eq,axiom,
    ! [X: nat,X6: set_nat] :
      ( ~ ( member_nat @ X @ X6 )
     => ( ( minus_minus_set_nat @ X6 @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_1722_Diff__insert__absorb,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A4 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1723_Diff__insert__absorb,axiom,
    ! [X: real,A4: set_real] :
      ( ~ ( member_real @ X @ A4 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A4 ) @ ( insert_real @ X @ bot_bot_set_real ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1724_Diff__insert__absorb,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1725_Diff__insert__absorb,axiom,
    ! [X: int,A4: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1726_Diff__insert__absorb,axiom,
    ! [X: $o,A4: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ ( insert_o @ X @ bot_bot_set_o ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1727_Diff__insert__absorb,axiom,
    ! [X: nat,A4: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1728_Diff__insert2,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1729_Diff__insert2,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B4 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1730_Diff__insert2,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1731_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1732_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A4 )
     => ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1733_insert__Diff,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1734_insert__Diff,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1735_insert__Diff,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1736_insert__Diff,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1737_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1738_Diff__insert,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Diff_insert
thf(fact_1739_Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B4 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ B4 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_1740_Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ B4 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_1741_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_1742_finite_OinsertI,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite5795047828879050333T_VEBT @ ( insert_VEBT_VEBT @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_1743_finite_OinsertI,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_o @ ( insert_o @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_1744_finite_OinsertI,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite_finite_nat @ ( insert_nat @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_1745_finite_OinsertI,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( finite_finite_int @ ( insert_int @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_1746_finite_OinsertI,axiom,
    ! [A4: set_complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_1747_finite_OinsertI,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite6017078050557962740nteger @ ( insert_Code_integer @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_1748_subset__insertI2,axiom,
    ! [A4: set_nat,B4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1749_subset__insertI2,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
     => ( ord_le4337996190870823476T_VEBT @ A4 @ ( insert_VEBT_VEBT @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1750_subset__insertI2,axiom,
    ! [A4: set_o,B4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1751_subset__insertI2,axiom,
    ! [A4: set_int,B4: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1752_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_1753_subset__insertI,axiom,
    ! [B4: set_VEBT_VEBT,A: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ B4 @ ( insert_VEBT_VEBT @ A @ B4 ) ) ).

% subset_insertI
thf(fact_1754_subset__insertI,axiom,
    ! [B4: set_o,A: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A @ B4 ) ) ).

% subset_insertI
thf(fact_1755_subset__insertI,axiom,
    ! [B4: set_int,A: int] : ( ord_less_eq_set_int @ B4 @ ( insert_int @ A @ B4 ) ) ).

% subset_insertI
thf(fact_1756_subset__insert,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B4 ) )
        = ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1757_subset__insert,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
        = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1758_subset__insert,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ B4 ) )
        = ( ord_le4337996190870823476T_VEBT @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1759_subset__insert,axiom,
    ! [X: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X @ A4 )
     => ( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X @ B4 ) )
        = ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1760_subset__insert,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ ( insert_set_nat @ X @ B4 ) )
        = ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1761_subset__insert,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B4 ) )
        = ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1762_insert__mono,axiom,
    ! [C3: set_nat,D4: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C3 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_1763_insert__mono,axiom,
    ! [C3: set_VEBT_VEBT,D4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ C3 @ D4 )
     => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ A @ C3 ) @ ( insert_VEBT_VEBT @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_1764_insert__mono,axiom,
    ! [C3: set_o,D4: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C3 @ D4 )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C3 ) @ ( insert_o @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_1765_insert__mono,axiom,
    ! [C3: set_int,D4: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C3 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C3 ) @ ( insert_int @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_1766_insert__subsetI,axiom,
    ! [X: $o,A4: set_o,X6: set_o] :
      ( ( member_o @ X @ A4 )
     => ( ( ord_less_eq_set_o @ X6 @ A4 )
       => ( ord_less_eq_set_o @ ( insert_o @ X @ X6 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1767_insert__subsetI,axiom,
    ! [X: nat,A4: set_nat,X6: set_nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( ord_less_eq_set_nat @ X6 @ A4 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X6 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1768_insert__subsetI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( ord_le4337996190870823476T_VEBT @ X6 @ A4 )
       => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X @ X6 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1769_insert__subsetI,axiom,
    ! [X: real,A4: set_real,X6: set_real] :
      ( ( member_real @ X @ A4 )
     => ( ( ord_less_eq_set_real @ X6 @ A4 )
       => ( ord_less_eq_set_real @ ( insert_real @ X @ X6 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1770_insert__subsetI,axiom,
    ! [X: set_nat,A4: set_set_nat,X6: set_set_nat] :
      ( ( member_set_nat @ X @ A4 )
     => ( ( ord_le6893508408891458716et_nat @ X6 @ A4 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ X6 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1771_insert__subsetI,axiom,
    ! [X: int,A4: set_int,X6: set_int] :
      ( ( member_int @ X @ A4 )
     => ( ( ord_less_eq_set_int @ X6 @ A4 )
       => ( ord_less_eq_set_int @ ( insert_int @ X @ X6 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1772_subset__Diff__insert,axiom,
    ! [A4: set_o,B4: set_o,X: $o,C3: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ ( insert_o @ X @ C3 ) ) )
      = ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ C3 ) )
        & ~ ( member_o @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1773_subset__Diff__insert,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,X: vEBT_VEBT,C3: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A4 @ ( minus_5127226145743854075T_VEBT @ B4 @ ( insert_VEBT_VEBT @ X @ C3 ) ) )
      = ( ( ord_le4337996190870823476T_VEBT @ A4 @ ( minus_5127226145743854075T_VEBT @ B4 @ C3 ) )
        & ~ ( member_VEBT_VEBT @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1774_subset__Diff__insert,axiom,
    ! [A4: set_real,B4: set_real,X: real,C3: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( minus_minus_set_real @ B4 @ ( insert_real @ X @ C3 ) ) )
      = ( ( ord_less_eq_set_real @ A4 @ ( minus_minus_set_real @ B4 @ C3 ) )
        & ~ ( member_real @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1775_subset__Diff__insert,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,X: set_nat,C3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( minus_2163939370556025621et_nat @ B4 @ ( insert_set_nat @ X @ C3 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A4 @ ( minus_2163939370556025621et_nat @ B4 @ C3 ) )
        & ~ ( member_set_nat @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1776_subset__Diff__insert,axiom,
    ! [A4: set_nat,B4: set_nat,X: nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C3 ) ) )
      = ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ C3 ) )
        & ~ ( member_nat @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1777_subset__Diff__insert,axiom,
    ! [A4: set_int,B4: set_int,X: int,C3: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( minus_minus_set_int @ B4 @ ( insert_int @ X @ C3 ) ) )
      = ( ( ord_less_eq_set_int @ A4 @ ( minus_minus_set_int @ B4 @ C3 ) )
        & ~ ( member_int @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1778_finite_Ocases,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A )
     => ( ( A != bot_bo8194388402131092736T_VEBT )
       => ~ ! [A5: set_VEBT_VEBT] :
              ( ? [A3: vEBT_VEBT] :
                  ( A
                  = ( insert_VEBT_VEBT @ A3 @ A5 ) )
             => ~ ( finite5795047828879050333T_VEBT @ A5 ) ) ) ) ).

% finite.cases
thf(fact_1779_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A5: set_complex] :
              ( ? [A3: complex] :
                  ( A
                  = ( insert_complex @ A3 @ A5 ) )
             => ~ ( finite3207457112153483333omplex @ A5 ) ) ) ) ).

% finite.cases
thf(fact_1780_finite_Ocases,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ~ ! [A5: set_Code_integer] :
              ( ? [A3: code_integer] :
                  ( A
                  = ( insert_Code_integer @ A3 @ A5 ) )
             => ~ ( finite6017078050557962740nteger @ A5 ) ) ) ) ).

% finite.cases
thf(fact_1781_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A5: set_nat] :
              ( ? [A3: nat] :
                  ( A
                  = ( insert_nat @ A3 @ A5 ) )
             => ~ ( finite_finite_nat @ A5 ) ) ) ) ).

% finite.cases
thf(fact_1782_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A5: set_int] :
              ( ? [A3: int] :
                  ( A
                  = ( insert_int @ A3 @ A5 ) )
             => ~ ( finite_finite_int @ A5 ) ) ) ) ).

% finite.cases
thf(fact_1783_finite_Ocases,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ~ ! [A5: set_o] :
              ( ? [A3: $o] :
                  ( A
                  = ( insert_o @ A3 @ A5 ) )
             => ~ ( finite_finite_o @ A5 ) ) ) ) ).

% finite.cases
thf(fact_1784_finite_Osimps,axiom,
    ( finite5795047828879050333T_VEBT
    = ( ^ [A7: set_VEBT_VEBT] :
          ( ( A7 = bot_bo8194388402131092736T_VEBT )
          | ? [A6: set_VEBT_VEBT,B7: vEBT_VEBT] :
              ( ( A7
                = ( insert_VEBT_VEBT @ B7 @ A6 ) )
              & ( finite5795047828879050333T_VEBT @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1785_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A7: set_complex] :
          ( ( A7 = bot_bot_set_complex )
          | ? [A6: set_complex,B7: complex] :
              ( ( A7
                = ( insert_complex @ B7 @ A6 ) )
              & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1786_finite_Osimps,axiom,
    ( finite6017078050557962740nteger
    = ( ^ [A7: set_Code_integer] :
          ( ( A7 = bot_bo3990330152332043303nteger )
          | ? [A6: set_Code_integer,B7: code_integer] :
              ( ( A7
                = ( insert_Code_integer @ B7 @ A6 ) )
              & ( finite6017078050557962740nteger @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1787_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A7: set_nat] :
          ( ( A7 = bot_bot_set_nat )
          | ? [A6: set_nat,B7: nat] :
              ( ( A7
                = ( insert_nat @ B7 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1788_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A7: set_int] :
          ( ( A7 = bot_bot_set_int )
          | ? [A6: set_int,B7: int] :
              ( ( A7
                = ( insert_int @ B7 @ A6 ) )
              & ( finite_finite_int @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1789_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A7: set_o] :
          ( ( A7 = bot_bot_set_o )
          | ? [A6: set_o,B7: $o] :
              ( ( A7
                = ( insert_o @ B7 @ A6 ) )
              & ( finite_finite_o @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1790_finite__induct,axiom,
    ! [F2: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F2 )
     => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,F3: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ F3 )
             => ( ~ ( member_VEBT_VEBT @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_VEBT_VEBT @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1791_finite__induct,axiom,
    ! [F2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X3: real,F3: set_real] :
              ( ( finite_finite_real @ F3 )
             => ( ~ ( member_real @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_real @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1792_finite__induct,axiom,
    ! [F2: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [X3: set_nat,F3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F3 )
             => ( ~ ( member_set_nat @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_set_nat @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1793_finite__induct,axiom,
    ! [F2: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X3: complex,F3: set_complex] :
              ( ( finite3207457112153483333omplex @ F3 )
             => ( ~ ( member_complex @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_complex @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1794_finite__induct,axiom,
    ! [F2: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F2 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,F3: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ F3 )
             => ( ~ ( member_Code_integer @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1795_finite__induct,axiom,
    ! [F2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1796_finite__induct,axiom,
    ! [F2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X3: int,F3: set_int] :
              ( ( finite_finite_int @ F3 )
             => ( ~ ( member_int @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_int @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1797_finite__induct,axiom,
    ! [F2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X3: $o,F3: set_o] :
              ( ( finite_finite_o @ F3 )
             => ( ~ ( member_o @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_o @ X3 @ F3 ) ) ) ) )
         => ( P2 @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1798_finite__ne__induct,axiom,
    ! [F2: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F2 )
     => ( ( F2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] : ( P2 @ ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) )
         => ( ! [X3: vEBT_VEBT,F3: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F3 )
               => ( ( F3 != bot_bo8194388402131092736T_VEBT )
                 => ( ~ ( member_VEBT_VEBT @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_VEBT_VEBT @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1799_finite__ne__induct,axiom,
    ! [F2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( F2 != bot_bot_set_real )
       => ( ! [X3: real] : ( P2 @ ( insert_real @ X3 @ bot_bot_set_real ) )
         => ( ! [X3: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( F3 != bot_bot_set_real )
                 => ( ~ ( member_real @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_real @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1800_finite__ne__induct,axiom,
    ! [F2: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( F2 != bot_bot_set_set_nat )
       => ( ! [X3: set_nat] : ( P2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
         => ( ! [X3: set_nat,F3: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F3 )
               => ( ( F3 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_set_nat @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1801_finite__ne__induct,axiom,
    ! [F2: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( F2 != bot_bot_set_complex )
       => ( ! [X3: complex] : ( P2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) )
         => ( ! [X3: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( F3 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_complex @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1802_finite__ne__induct,axiom,
    ! [F2: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F2 )
     => ( ( F2 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] : ( P2 @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) )
         => ( ! [X3: code_integer,F3: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F3 )
               => ( ( F3 != bot_bo3990330152332043303nteger )
                 => ( ~ ( member_Code_integer @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1803_finite__ne__induct,axiom,
    ! [F2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X3: nat] : ( P2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
         => ( ! [X3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1804_finite__ne__induct,axiom,
    ! [F2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( F2 != bot_bot_set_int )
       => ( ! [X3: int] : ( P2 @ ( insert_int @ X3 @ bot_bot_set_int ) )
         => ( ! [X3: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( F3 != bot_bot_set_int )
                 => ( ~ ( member_int @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_int @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1805_finite__ne__induct,axiom,
    ! [F2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X3: $o] : ( P2 @ ( insert_o @ X3 @ bot_bot_set_o ) )
         => ( ! [X3: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ ( member_o @ X3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_o @ X3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1806_infinite__finite__induct,axiom,
    ! [P2: set_VEBT_VEBT > $o,A4: set_VEBT_VEBT] :
      ( ! [A5: set_VEBT_VEBT] :
          ( ~ ( finite5795047828879050333T_VEBT @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,F3: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ F3 )
             => ( ~ ( member_VEBT_VEBT @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_VEBT_VEBT @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1807_infinite__finite__induct,axiom,
    ! [P2: set_real > $o,A4: set_real] :
      ( ! [A5: set_real] :
          ( ~ ( finite_finite_real @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X3: real,F3: set_real] :
              ( ( finite_finite_real @ F3 )
             => ( ~ ( member_real @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_real @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1808_infinite__finite__induct,axiom,
    ! [P2: set_set_nat > $o,A4: set_set_nat] :
      ( ! [A5: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [X3: set_nat,F3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F3 )
             => ( ~ ( member_set_nat @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_set_nat @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1809_infinite__finite__induct,axiom,
    ! [P2: set_complex > $o,A4: set_complex] :
      ( ! [A5: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X3: complex,F3: set_complex] :
              ( ( finite3207457112153483333omplex @ F3 )
             => ( ~ ( member_complex @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_complex @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1810_infinite__finite__induct,axiom,
    ! [P2: set_Code_integer > $o,A4: set_Code_integer] :
      ( ! [A5: set_Code_integer] :
          ( ~ ( finite6017078050557962740nteger @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,F3: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ F3 )
             => ( ~ ( member_Code_integer @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1811_infinite__finite__induct,axiom,
    ! [P2: set_nat > $o,A4: set_nat] :
      ( ! [A5: set_nat] :
          ( ~ ( finite_finite_nat @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1812_infinite__finite__induct,axiom,
    ! [P2: set_int > $o,A4: set_int] :
      ( ! [A5: set_int] :
          ( ~ ( finite_finite_int @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X3: int,F3: set_int] :
              ( ( finite_finite_int @ F3 )
             => ( ~ ( member_int @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_int @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1813_infinite__finite__induct,axiom,
    ! [P2: set_o > $o,A4: set_o] :
      ( ! [A5: set_o] :
          ( ~ ( finite_finite_o @ A5 )
         => ( P2 @ A5 ) )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X3: $o,F3: set_o] :
              ( ( finite_finite_o @ F3 )
             => ( ~ ( member_o @ X3 @ F3 )
               => ( ( P2 @ F3 )
                 => ( P2 @ ( insert_o @ X3 @ F3 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1814_infinite__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ~ ( finite5795047828879050333T_VEBT @ S3 )
     => ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% infinite_remove
thf(fact_1815_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_1816_infinite__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S3 )
     => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ).

% infinite_remove
thf(fact_1817_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ ( finite_finite_int @ S3 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_1818_infinite__remove,axiom,
    ! [S3: set_o,A: $o] :
      ( ~ ( finite_finite_o @ S3 )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_1819_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_1820_infinite__coinduct,axiom,
    ! [X6: set_VEBT_VEBT > $o,A4: set_VEBT_VEBT] :
      ( ( X6 @ A4 )
     => ( ! [A5: set_VEBT_VEBT] :
            ( ( X6 @ A5 )
           => ? [X5: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X5 @ A5 )
                & ( ( X6 @ ( minus_5127226145743854075T_VEBT @ A5 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) )
                  | ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A5 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
       => ~ ( finite5795047828879050333T_VEBT @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1821_infinite__coinduct,axiom,
    ! [X6: set_complex > $o,A4: set_complex] :
      ( ( X6 @ A4 )
     => ( ! [A5: set_complex] :
            ( ( X6 @ A5 )
           => ? [X5: complex] :
                ( ( member_complex @ X5 @ A5 )
                & ( ( X6 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1822_infinite__coinduct,axiom,
    ! [X6: set_Code_integer > $o,A4: set_Code_integer] :
      ( ( X6 @ A4 )
     => ( ! [A5: set_Code_integer] :
            ( ( X6 @ A5 )
           => ? [X5: code_integer] :
                ( ( member_Code_integer @ X5 @ A5 )
                & ( ( X6 @ ( minus_2355218937544613996nteger @ A5 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) )
                  | ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A5 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
       => ~ ( finite6017078050557962740nteger @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1823_infinite__coinduct,axiom,
    ! [X6: set_int > $o,A4: set_int] :
      ( ( X6 @ A4 )
     => ( ! [A5: set_int] :
            ( ( X6 @ A5 )
           => ? [X5: int] :
                ( ( member_int @ X5 @ A5 )
                & ( ( X6 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X5 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A5 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1824_infinite__coinduct,axiom,
    ! [X6: set_o > $o,A4: set_o] :
      ( ( X6 @ A4 )
     => ( ! [A5: set_o] :
            ( ( X6 @ A5 )
           => ? [X5: $o] :
                ( ( member_o @ X5 @ A5 )
                & ( ( X6 @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1825_infinite__coinduct,axiom,
    ! [X6: set_nat > $o,A4: set_nat] :
      ( ( X6 @ A4 )
     => ( ! [A5: set_nat] :
            ( ( X6 @ A5 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ A5 )
                & ( ( X6 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1826_finite__empty__induct,axiom,
    ! [A4: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: vEBT_VEBT,A5: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A5 )
             => ( ( member_VEBT_VEBT @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_5127226145743854075T_VEBT @ A5 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
         => ( P2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% finite_empty_induct
thf(fact_1827_finite__empty__induct,axiom,
    ! [A4: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: real,A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ( member_real @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_minus_set_real @ A5 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
         => ( P2 @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_1828_finite__empty__induct,axiom,
    ! [A4: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: set_nat,A5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A5 )
             => ( ( member_set_nat @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P2 @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_1829_finite__empty__induct,axiom,
    ! [A4: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: complex,A5: set_complex] :
              ( ( finite3207457112153483333omplex @ A5 )
             => ( ( member_complex @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P2 @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_1830_finite__empty__induct,axiom,
    ! [A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: code_integer,A5: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A5 )
             => ( ( member_Code_integer @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_2355218937544613996nteger @ A5 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
         => ( P2 @ bot_bo3990330152332043303nteger ) ) ) ) ).

% finite_empty_induct
thf(fact_1831_finite__empty__induct,axiom,
    ! [A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: int,A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ( member_int @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_minus_set_int @ A5 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
         => ( P2 @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_1832_finite__empty__induct,axiom,
    ! [A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: $o,A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ( member_o @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_minus_set_o @ A5 @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ) )
         => ( P2 @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_1833_finite__empty__induct,axiom,
    ! [A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P2 @ A4 )
       => ( ! [A3: nat,A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ( member_nat @ A3 @ A5 )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P2 @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_1834_subset__singleton__iff,axiom,
    ! [X6: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ X6 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( X6 = bot_bo8194388402131092736T_VEBT )
        | ( X6
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singleton_iff
thf(fact_1835_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X6 = bot_bot_set_nat )
        | ( X6
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_1836_subset__singleton__iff,axiom,
    ! [X6: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X6 = bot_bot_set_o )
        | ( X6
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_1837_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X6 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X6 = bot_bot_set_int )
        | ( X6
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_1838_subset__singletonD,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
     => ( ( A4 = bot_bo8194388402131092736T_VEBT )
        | ( A4
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singletonD
thf(fact_1839_subset__singletonD,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A4 = bot_bot_set_nat )
        | ( A4
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_1840_subset__singletonD,axiom,
    ! [A4: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A4 = bot_bot_set_o )
        | ( A4
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_1841_subset__singletonD,axiom,
    ! [A4: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A4 = bot_bot_set_int )
        | ( A4
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_1842_subset__insert__iff,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ B4 ) )
      = ( ( ( member_VEBT_VEBT @ X @ A4 )
         => ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A4 )
         => ( ord_le4337996190870823476T_VEBT @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1843_subset__insert__iff,axiom,
    ! [A4: set_real,X: real,B4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X @ B4 ) )
      = ( ( ( member_real @ X @ A4 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B4 ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1844_subset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( insert_set_nat @ X @ B4 ) )
      = ( ( ( member_set_nat @ X @ A4 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B4 ) )
        & ( ~ ( member_set_nat @ X @ A4 )
         => ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1845_subset__insert__iff,axiom,
    ! [A4: set_o,X: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B4 ) )
      = ( ( ( member_o @ X @ A4 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1846_subset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
      = ( ( ( member_nat @ X @ A4 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1847_subset__insert__iff,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B4 ) )
      = ( ( ( member_int @ X @ A4 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1848_Diff__single__insert,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B4 )
     => ( ord_le4337996190870823476T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1849_Diff__single__insert,axiom,
    ! [A4: set_o,X: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1850_Diff__single__insert,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1851_Diff__single__insert,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 )
     => ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1852_remove__subset,axiom,
    ! [X: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ S3 )
     => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ S3 ) ) ).

% remove_subset
thf(fact_1853_remove__subset,axiom,
    ! [X: real,S3: set_real] :
      ( ( member_real @ X @ S3 )
     => ( ord_less_set_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ X @ bot_bot_set_real ) ) @ S3 ) ) ).

% remove_subset
thf(fact_1854_remove__subset,axiom,
    ! [X: set_nat,S3: set_set_nat] :
      ( ( member_set_nat @ X @ S3 )
     => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ S3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ S3 ) ) ).

% remove_subset
thf(fact_1855_remove__subset,axiom,
    ! [X: int,S3: set_int] :
      ( ( member_int @ X @ S3 )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ X @ bot_bot_set_int ) ) @ S3 ) ) ).

% remove_subset
thf(fact_1856_remove__subset,axiom,
    ! [X: $o,S3: set_o] :
      ( ( member_o @ X @ S3 )
     => ( ord_less_set_o @ ( minus_minus_set_o @ S3 @ ( insert_o @ X @ bot_bot_set_o ) ) @ S3 ) ) ).

% remove_subset
thf(fact_1857_remove__subset,axiom,
    ! [X: nat,S3: set_nat] :
      ( ( member_nat @ X @ S3 )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S3 ) ) ).

% remove_subset
thf(fact_1858_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S5 )
             => ( ! [Y5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y5 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_VEBT_VEBT @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1859_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X3: real,S5: set_real] :
              ( ( finite_finite_real @ S5 )
             => ( ! [Y5: real] :
                    ( ( member_real @ Y5 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_real @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1860_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X3: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_complex @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1861_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P2: set_Code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S5: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S5 )
             => ( ! [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_Code_integer @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1862_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,S5: set_nat] :
              ( ( finite_finite_nat @ S5 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_nat @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1863_finite__ranking__induct,axiom,
    ! [S3: set_int,P2: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X3: int,S5: set_int] :
              ( ( finite_finite_int @ S5 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_int @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1864_finite__ranking__induct,axiom,
    ! [S3: set_o,P2: set_o > $o,F: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X3: $o,S5: set_o] :
              ( ( finite_finite_o @ S5 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_o @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1865_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o,F: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S5 )
             => ( ! [Y5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y5 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_VEBT_VEBT @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1866_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X3: real,S5: set_real] :
              ( ( finite_finite_real @ S5 )
             => ( ! [Y5: real] :
                    ( ( member_real @ Y5 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_real @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1867_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X3: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S5 )
                 => ( P2 @ ( insert_complex @ X3 @ S5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1868_finite__linorder__max__induct,axiom,
    ! [A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A5: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A5 )
             => ( ! [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ A5 )
                   => ( ord_le6747313008572928689nteger @ X5 @ B3 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_Code_integer @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1869_finite__linorder__max__induct,axiom,
    ! [A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [B3: $o,A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ! [X5: $o] :
                    ( ( member_o @ X5 @ A5 )
                   => ( ord_less_o @ X5 @ B3 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_o @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1870_finite__linorder__max__induct,axiom,
    ! [A4: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [B3: real,A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A5 )
                   => ( ord_less_real @ X5 @ B3 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_real @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1871_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,P2: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P2 @ bot_bot_set_rat )
       => ( ! [B3: rat,A5: set_rat] :
              ( ( finite_finite_rat @ A5 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A5 )
                   => ( ord_less_rat @ X5 @ B3 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_rat @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1872_finite__linorder__max__induct,axiom,
    ! [A4: set_num,P2: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P2 @ bot_bot_set_num )
       => ( ! [B3: num,A5: set_num] :
              ( ( finite_finite_num @ A5 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A5 )
                   => ( ord_less_num @ X5 @ B3 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_num @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1873_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [B3: nat,A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A5 )
                   => ( ord_less_nat @ X5 @ B3 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_nat @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1874_finite__linorder__max__induct,axiom,
    ! [A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [B3: int,A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A5 )
                   => ( ord_less_int @ X5 @ B3 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_int @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1875_finite__linorder__min__induct,axiom,
    ! [A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A5: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A5 )
             => ( ! [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ A5 )
                   => ( ord_le6747313008572928689nteger @ B3 @ X5 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_Code_integer @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1876_finite__linorder__min__induct,axiom,
    ! [A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [B3: $o,A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ! [X5: $o] :
                    ( ( member_o @ X5 @ A5 )
                   => ( ord_less_o @ B3 @ X5 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_o @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1877_finite__linorder__min__induct,axiom,
    ! [A4: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [B3: real,A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A5 )
                   => ( ord_less_real @ B3 @ X5 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_real @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1878_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,P2: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P2 @ bot_bot_set_rat )
       => ( ! [B3: rat,A5: set_rat] :
              ( ( finite_finite_rat @ A5 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A5 )
                   => ( ord_less_rat @ B3 @ X5 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_rat @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1879_finite__linorder__min__induct,axiom,
    ! [A4: set_num,P2: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P2 @ bot_bot_set_num )
       => ( ! [B3: num,A5: set_num] :
              ( ( finite_finite_num @ A5 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A5 )
                   => ( ord_less_num @ B3 @ X5 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_num @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1880_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [B3: nat,A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A5 )
                   => ( ord_less_nat @ B3 @ X5 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_nat @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1881_finite__linorder__min__induct,axiom,
    ! [A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [B3: int,A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A5 )
                   => ( ord_less_int @ B3 @ X5 ) )
               => ( ( P2 @ A5 )
                 => ( P2 @ ( insert_int @ B3 @ A5 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1882_finite__subset__induct,axiom,
    ! [F2: set_VEBT_VEBT,A4: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F2 )
     => ( ( ord_le4337996190870823476T_VEBT @ F2 @ A4 )
       => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F3: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F3 )
               => ( ( member_VEBT_VEBT @ A3 @ A4 )
                 => ( ~ ( member_VEBT_VEBT @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_VEBT_VEBT @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1883_finite__subset__induct,axiom,
    ! [F2: set_real,A4: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( ord_less_eq_set_real @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_real )
         => ( ! [A3: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( member_real @ A3 @ A4 )
                 => ( ~ ( member_real @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_real @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1884_finite__subset__induct,axiom,
    ! [F2: set_set_nat,A4: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( ord_le6893508408891458716et_nat @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A3: set_nat,F3: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F3 )
               => ( ( member_set_nat @ A3 @ A4 )
                 => ( ~ ( member_set_nat @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_set_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1885_finite__subset__induct,axiom,
    ! [F2: set_complex,A4: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ord_le211207098394363844omplex @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_complex )
         => ( ! [A3: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( member_complex @ A3 @ A4 )
                 => ( ~ ( member_complex @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_complex @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1886_finite__subset__induct,axiom,
    ! [F2: set_Code_integer,A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F2 )
     => ( ( ord_le7084787975880047091nteger @ F2 @ A4 )
       => ( ( P2 @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F3: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F3 )
               => ( ( member_Code_integer @ A3 @ A4 )
                 => ( ~ ( member_Code_integer @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_Code_integer @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1887_finite__subset__induct,axiom,
    ! [F2: set_nat,A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A4 )
                 => ( ~ ( member_nat @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1888_finite__subset__induct,axiom,
    ! [F2: set_o,A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_o )
         => ( ! [A3: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( member_o @ A3 @ A4 )
                 => ( ~ ( member_o @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_o @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1889_finite__subset__induct,axiom,
    ! [F2: set_int,A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( ord_less_eq_set_int @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_int )
         => ( ! [A3: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( member_int @ A3 @ A4 )
                 => ( ~ ( member_int @ A3 @ F3 )
                   => ( ( P2 @ F3 )
                     => ( P2 @ ( insert_int @ A3 @ F3 ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1890_finite__subset__induct_H,axiom,
    ! [F2: set_VEBT_VEBT,A4: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F2 )
     => ( ( ord_le4337996190870823476T_VEBT @ F2 @ A4 )
       => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F3: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F3 )
               => ( ( member_VEBT_VEBT @ A3 @ A4 )
                 => ( ( ord_le4337996190870823476T_VEBT @ F3 @ A4 )
                   => ( ~ ( member_VEBT_VEBT @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_VEBT_VEBT @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1891_finite__subset__induct_H,axiom,
    ! [F2: set_real,A4: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( ord_less_eq_set_real @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_real )
         => ( ! [A3: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( member_real @ A3 @ A4 )
                 => ( ( ord_less_eq_set_real @ F3 @ A4 )
                   => ( ~ ( member_real @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_real @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1892_finite__subset__induct_H,axiom,
    ! [F2: set_set_nat,A4: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( ord_le6893508408891458716et_nat @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A3: set_nat,F3: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F3 )
               => ( ( member_set_nat @ A3 @ A4 )
                 => ( ( ord_le6893508408891458716et_nat @ F3 @ A4 )
                   => ( ~ ( member_set_nat @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_set_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1893_finite__subset__induct_H,axiom,
    ! [F2: set_complex,A4: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ord_le211207098394363844omplex @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_complex )
         => ( ! [A3: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( member_complex @ A3 @ A4 )
                 => ( ( ord_le211207098394363844omplex @ F3 @ A4 )
                   => ( ~ ( member_complex @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_complex @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1894_finite__subset__induct_H,axiom,
    ! [F2: set_Code_integer,A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F2 )
     => ( ( ord_le7084787975880047091nteger @ F2 @ A4 )
       => ( ( P2 @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F3: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F3 )
               => ( ( member_Code_integer @ A3 @ A4 )
                 => ( ( ord_le7084787975880047091nteger @ F3 @ A4 )
                   => ( ~ ( member_Code_integer @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_Code_integer @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1895_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A4 )
                 => ( ( ord_less_eq_set_nat @ F3 @ A4 )
                   => ( ~ ( member_nat @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1896_finite__subset__induct_H,axiom,
    ! [F2: set_o,A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_o )
         => ( ! [A3: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( member_o @ A3 @ A4 )
                 => ( ( ord_less_eq_set_o @ F3 @ A4 )
                   => ( ~ ( member_o @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_o @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1897_finite__subset__induct_H,axiom,
    ! [F2: set_int,A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( ord_less_eq_set_int @ F2 @ A4 )
       => ( ( P2 @ bot_bot_set_int )
         => ( ! [A3: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( member_int @ A3 @ A4 )
                 => ( ( ord_less_eq_set_int @ F3 @ A4 )
                   => ( ~ ( member_int @ A3 @ F3 )
                     => ( ( P2 @ F3 )
                       => ( P2 @ ( insert_int @ A3 @ F3 ) ) ) ) ) ) )
           => ( P2 @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1898_remove__induct,axiom,
    ! [P2: set_VEBT_VEBT > $o,B4: set_VEBT_VEBT] :
      ( ( P2 @ bot_bo8194388402131092736T_VEBT )
     => ( ( ~ ( finite5795047828879050333T_VEBT @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A5 )
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A5 @ B4 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ A5 )
                       => ( P2 @ ( minus_5127226145743854075T_VEBT @ A5 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1899_remove__induct,axiom,
    ! [P2: set_real > $o,B4: set_real] :
      ( ( P2 @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ( A5 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A5 @ B4 )
                 => ( ! [X5: real] :
                        ( ( member_real @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_real @ A5 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1900_remove__induct,axiom,
    ! [P2: set_set_nat > $o,B4: set_set_nat] :
      ( ( P2 @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A5 )
             => ( ( A5 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A5 @ B4 )
                 => ( ! [X5: set_nat] :
                        ( ( member_set_nat @ X5 @ A5 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1901_remove__induct,axiom,
    ! [P2: set_complex > $o,B4: set_complex] :
      ( ( P2 @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_complex] :
              ( ( finite3207457112153483333omplex @ A5 )
             => ( ( A5 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A5 @ B4 )
                 => ( ! [X5: complex] :
                        ( ( member_complex @ X5 @ A5 )
                       => ( P2 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1902_remove__induct,axiom,
    ! [P2: set_Code_integer > $o,B4: set_Code_integer] :
      ( ( P2 @ bot_bo3990330152332043303nteger )
     => ( ( ~ ( finite6017078050557962740nteger @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A5 )
             => ( ( A5 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A5 @ B4 )
                 => ( ! [X5: code_integer] :
                        ( ( member_Code_integer @ X5 @ A5 )
                       => ( P2 @ ( minus_2355218937544613996nteger @ A5 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1903_remove__induct,axiom,
    ! [P2: set_o > $o,B4: set_o] :
      ( ( P2 @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ( A5 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A5 @ B4 )
                 => ( ! [X5: $o] :
                        ( ( member_o @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1904_remove__induct,axiom,
    ! [P2: set_nat > $o,B4: set_nat] :
      ( ( P2 @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ( A5 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A5 @ B4 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1905_remove__induct,axiom,
    ! [P2: set_int > $o,B4: set_int] :
      ( ( P2 @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B4 )
         => ( P2 @ B4 ) )
       => ( ! [A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ( A5 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A5 @ B4 )
                 => ( ! [X5: int] :
                        ( ( member_int @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1906_finite__remove__induct,axiom,
    ! [B4: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
       => ( ! [A5: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A5 )
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A5 @ B4 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ A5 )
                       => ( P2 @ ( minus_5127226145743854075T_VEBT @ A5 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1907_finite__remove__induct,axiom,
    ! [B4: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ B4 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ( A5 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A5 @ B4 )
                 => ( ! [X5: real] :
                        ( ( member_real @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_real @ A5 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1908_finite__remove__induct,axiom,
    ! [B4: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [A5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A5 )
             => ( ( A5 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A5 @ B4 )
                 => ( ! [X5: set_nat] :
                        ( ( member_set_nat @ X5 @ A5 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1909_finite__remove__induct,axiom,
    ! [B4: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [A5: set_complex] :
              ( ( finite3207457112153483333omplex @ A5 )
             => ( ( A5 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A5 @ B4 )
                 => ( ! [X5: complex] :
                        ( ( member_complex @ X5 @ A5 )
                       => ( P2 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1910_finite__remove__induct,axiom,
    ! [B4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [A5: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A5 )
             => ( ( A5 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A5 @ B4 )
                 => ( ! [X5: code_integer] :
                        ( ( member_Code_integer @ X5 @ A5 )
                       => ( P2 @ ( minus_2355218937544613996nteger @ A5 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1911_finite__remove__induct,axiom,
    ! [B4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ B4 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ( A5 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A5 @ B4 )
                 => ( ! [X5: $o] :
                        ( ( member_o @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1912_finite__remove__induct,axiom,
    ! [B4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ( A5 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A5 @ B4 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1913_finite__remove__induct,axiom,
    ! [B4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ B4 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ( A5 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A5 @ B4 )
                 => ( ! [X5: int] :
                        ( ( member_int @ X5 @ A5 )
                       => ( P2 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
                   => ( P2 @ A5 ) ) ) ) )
         => ( P2 @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1914_finite__induct__select,axiom,
    ! [S3: set_VEBT_VEBT,P2: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P2 @ bot_bo8194388402131092736T_VEBT )
       => ( ! [T4: set_VEBT_VEBT] :
              ( ( ord_le3480810397992357184T_VEBT @ T4 @ S3 )
             => ( ( P2 @ T4 )
               => ? [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( minus_5127226145743854075T_VEBT @ S3 @ T4 ) )
                    & ( P2 @ ( insert_VEBT_VEBT @ X5 @ T4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_1915_finite__induct__select,axiom,
    ! [S3: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [T4: set_complex] :
              ( ( ord_less_set_complex @ T4 @ S3 )
             => ( ( P2 @ T4 )
               => ? [X5: complex] :
                    ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ S3 @ T4 ) )
                    & ( P2 @ ( insert_complex @ X5 @ T4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_1916_finite__induct__select,axiom,
    ! [S3: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [T4: set_Code_integer] :
              ( ( ord_le1307284697595431911nteger @ T4 @ S3 )
             => ( ( P2 @ T4 )
               => ? [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ ( minus_2355218937544613996nteger @ S3 @ T4 ) )
                    & ( P2 @ ( insert_Code_integer @ X5 @ T4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_1917_finite__induct__select,axiom,
    ! [S3: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ S3 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [T4: set_int] :
              ( ( ord_less_set_int @ T4 @ S3 )
             => ( ( P2 @ T4 )
               => ? [X5: int] :
                    ( ( member_int @ X5 @ ( minus_minus_set_int @ S3 @ T4 ) )
                    & ( P2 @ ( insert_int @ X5 @ T4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_1918_finite__induct__select,axiom,
    ! [S3: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ S3 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [T4: set_o] :
              ( ( ord_less_set_o @ T4 @ S3 )
             => ( ( P2 @ T4 )
               => ? [X5: $o] :
                    ( ( member_o @ X5 @ ( minus_minus_set_o @ S3 @ T4 ) )
                    & ( P2 @ ( insert_o @ X5 @ T4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_1919_finite__induct__select,axiom,
    ! [S3: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [T4: set_nat] :
              ( ( ord_less_set_nat @ T4 @ S3 )
             => ( ( P2 @ T4 )
               => ? [X5: nat] :
                    ( ( member_nat @ X5 @ ( minus_minus_set_nat @ S3 @ T4 ) )
                    & ( P2 @ ( insert_nat @ X5 @ T4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_1920_psubset__insert__iff,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ B4 ) )
      = ( ( ( member_VEBT_VEBT @ X @ B4 )
         => ( ord_le3480810397992357184T_VEBT @ A4 @ B4 ) )
        & ( ~ ( member_VEBT_VEBT @ X @ B4 )
         => ( ( ( member_VEBT_VEBT @ X @ A4 )
             => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) )
            & ( ~ ( member_VEBT_VEBT @ X @ A4 )
             => ( ord_le4337996190870823476T_VEBT @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1921_psubset__insert__iff,axiom,
    ! [A4: set_real,X: real,B4: set_real] :
      ( ( ord_less_set_real @ A4 @ ( insert_real @ X @ B4 ) )
      = ( ( ( member_real @ X @ B4 )
         => ( ord_less_set_real @ A4 @ B4 ) )
        & ( ~ ( member_real @ X @ B4 )
         => ( ( ( member_real @ X @ A4 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B4 ) )
            & ( ~ ( member_real @ X @ A4 )
             => ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1922_psubset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B4: set_set_nat] :
      ( ( ord_less_set_set_nat @ A4 @ ( insert_set_nat @ X @ B4 ) )
      = ( ( ( member_set_nat @ X @ B4 )
         => ( ord_less_set_set_nat @ A4 @ B4 ) )
        & ( ~ ( member_set_nat @ X @ B4 )
         => ( ( ( member_set_nat @ X @ A4 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B4 ) )
            & ( ~ ( member_set_nat @ X @ A4 )
             => ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1923_psubset__insert__iff,axiom,
    ! [A4: set_o,X: $o,B4: set_o] :
      ( ( ord_less_set_o @ A4 @ ( insert_o @ X @ B4 ) )
      = ( ( ( member_o @ X @ B4 )
         => ( ord_less_set_o @ A4 @ B4 ) )
        & ( ~ ( member_o @ X @ B4 )
         => ( ( ( member_o @ X @ A4 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
            & ( ~ ( member_o @ X @ A4 )
             => ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1924_psubset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
      = ( ( ( member_nat @ X @ B4 )
         => ( ord_less_set_nat @ A4 @ B4 ) )
        & ( ~ ( member_nat @ X @ B4 )
         => ( ( ( member_nat @ X @ A4 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
            & ( ~ ( member_nat @ X @ A4 )
             => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1925_psubset__insert__iff,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( ( ord_less_set_int @ A4 @ ( insert_int @ X @ B4 ) )
      = ( ( ( member_int @ X @ B4 )
         => ( ord_less_set_int @ A4 @ B4 ) )
        & ( ~ ( member_int @ X @ B4 )
         => ( ( ( member_int @ X @ A4 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
            & ( ~ ( member_int @ X @ A4 )
             => ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1926_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_1927_finite__list,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ? [Xs3: list_real] :
          ( ( set_real2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_1928_finite__list,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ? [Xs3: list_o] :
          ( ( set_o2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_1929_finite__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_1930_finite__list,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_1931_finite__list,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_1932_finite__list,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ? [Xs3: list_Code_integer] :
          ( ( set_Code_integer2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_1933_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B4 )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1934_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B4 )
      = ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1935_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1936_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B4 )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
           => ( member_real @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1937_subset__code_I1_J,axiom,
    ! [Xs: list_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B4 )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
           => ( member_o @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1938_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B4 )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
           => ( member_int @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1939_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_1940_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_1941_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_1942_set__vebt__pred,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% set_vebt_pred
thf(fact_1943_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_1944_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_1945_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_1946_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_1947_set__vebt__maxt,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
        = ( vEBT_VEBT_max_in_set @ ( vEBT_set_vebt @ T ) @ X ) ) ) ).

% set_vebt_maxt
thf(fact_1948_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_1949_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_1950_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_1951_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_1952_vebt__minNull__mint,axiom,
    ( vEBT_VEBT_minNull
    = ( ^ [T2: vEBT_VEBT] :
          ( ( vEBT_vebt_mint @ T2 )
          = none_nat ) ) ) ).

% vebt_minNull_mint
thf(fact_1953_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_1954_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_1955_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_1956_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_1957_set__vebt__mint_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
        = ( ( member_nat @ X @ ( vEBT_set_vebt @ T ) )
          & ! [X4: nat] :
              ( ( member_nat @ X4 @ ( vEBT_set_vebt @ T ) )
             => ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ).

% set_vebt_mint'
thf(fact_1958_set__vebt__maxt_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
        = ( ( member_nat @ X @ ( vEBT_set_vebt @ T ) )
          & ! [X4: nat] :
              ( ( member_nat @ X4 @ ( vEBT_set_vebt @ T ) )
             => ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ).

% set_vebt_maxt'
thf(fact_1959_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_1960_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_1961_set__vebt__mint,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
        = ( vEBT_VEBT_min_in_set @ ( vEBT_set_vebt @ T ) @ X ) ) ) ).

% set_vebt_mint
thf(fact_1962_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_1963_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_1964_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_1965_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz ) @ Va2 )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_1966_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz @ Va2 ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_1967_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd2 ) @ Ve2 )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_1968_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_1969_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_1970_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_1971_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu2: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_1972_vebt__succ__code_I1_J,axiom,
    ! [B: $o,X: nat,A: $o] :
      ( ( ( B
          & ( X = zero_zero_nat ) )
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ ( B
            & ( X = zero_zero_nat ) )
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ A @ B ) @ X )
          = none_nat ) ) ) ).

% vebt_succ_code(1)
thf(fact_1973_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_1974_vebt__pred__code,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ X )
          = none_nat ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( A
               => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ X )
                  = ( some_nat @ zero_zero_nat ) ) )
              & ( ~ A
               => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ X )
                  = none_nat ) ) ) )
          & ( ( X != one_one_nat )
           => ( ( B
               => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ X )
                  = ( some_nat @ one_one_nat ) ) )
              & ( ~ B
               => ( ( A
                   => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ X )
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A
                   => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ X )
                      = none_nat ) ) ) ) ) ) ) ) ) ).

% vebt_pred_code
thf(fact_1975_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va2: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_1976_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_1977_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_1978_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_1979_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_1980_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_1981_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_1982_set__vebt__mint_H_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( ( vEBT_set_vebt @ T )
            = bot_bot_set_nat )
         => ( ( vEBT_vebt_mint @ T )
            = none_nat ) )
        & ( ( ( vEBT_set_vebt @ T )
           != bot_bot_set_nat )
         => ( ( vEBT_vebt_mint @ T )
            = ( some_nat @ ( lattic8721135487736765967in_nat @ ( vEBT_set_vebt @ T ) ) ) ) ) ) ) ).

% set_vebt_mint''
thf(fact_1983_set__vebt__maxt_H_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( ( vEBT_set_vebt @ T )
            = bot_bot_set_nat )
         => ( ( vEBT_vebt_maxt @ T )
            = none_nat ) )
        & ( ( ( vEBT_set_vebt @ T )
           != bot_bot_set_nat )
         => ( ( vEBT_vebt_maxt @ T )
            = ( some_nat @ ( lattic8265883725875713057ax_nat @ ( vEBT_set_vebt @ T ) ) ) ) ) ) ) ).

% set_vebt_maxt''
thf(fact_1984_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_1985_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_1986_Max__singleton,axiom,
    ! [X: int] :
      ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% Max_singleton
thf(fact_1987_Max__singleton,axiom,
    ! [X: $o] :
      ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Max_singleton
thf(fact_1988_Max__singleton,axiom,
    ! [X: nat] :
      ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% Max_singleton
thf(fact_1989_Min__singleton,axiom,
    ! [X: int] :
      ( ( lattic8718645017227715691in_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% Min_singleton
thf(fact_1990_Min__singleton,axiom,
    ! [X: $o] :
      ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Min_singleton
thf(fact_1991_Min__singleton,axiom,
    ! [X: nat] :
      ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% Min_singleton
thf(fact_1992_Max_Obounded__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A4 ) @ X )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le3102999989581377725nteger @ X4 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1993_Max_Obounded__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A4 ) @ X )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_eq_o @ X4 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1994_Max_Obounded__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A4 ) @ X )
          = ( ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_eq_rat @ X4 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1995_Max_Obounded__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A4 ) @ X )
          = ( ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_eq_num @ X4 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1996_Max_Obounded__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_eq_int @ X4 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1997_Max_Obounded__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1998_Max__less__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le6747313008572928689nteger @ ( lattic4901227151466704046nteger @ A4 ) @ X )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le6747313008572928689nteger @ X4 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_1999_Max__less__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_o @ ( lattic1921953407002678535_Max_o @ A4 ) @ X )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_o @ X4 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_2000_Max__less__iff,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ord_less_real @ ( lattic4275903605611617917x_real @ A4 ) @ X )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ord_less_real @ X4 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_2001_Max__less__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_rat @ ( lattic7630753665789217321ax_rat @ A4 ) @ X )
          = ( ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_rat @ X4 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_2002_Max__less__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_num @ ( lattic4823215512031491691ax_num @ A4 ) @ X )
          = ( ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_num @ X4 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_2003_Max__less__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_int @ X4 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_2004_Max__less__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_nat @ X4 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_2005_Min_Obounded__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A4 ) )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le3102999989581377725nteger @ X @ X4 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_2006_Min_Obounded__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A4 ) )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_eq_o @ X @ X4 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_2007_Min_Obounded__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A4 ) )
          = ( ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_eq_rat @ X @ X4 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_2008_Min_Obounded__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ X @ ( lattic5278467273892544601in_num @ A4 ) )
          = ( ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_eq_num @ X @ X4 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_2009_Min_Obounded__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A4 ) )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_eq_int @ X @ X4 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_2010_Min_Obounded__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A4 ) )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_2011_Min__gr__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le6747313008572928689nteger @ X @ ( lattic1063845414844153500nteger @ A4 ) )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le6747313008572928689nteger @ X @ X4 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_2012_Min__gr__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_o @ X @ ( lattic1973801136483472281_Min_o @ A4 ) )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_o @ X @ X4 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_2013_Min__gr__iff,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ord_less_real @ X @ ( lattic3629708407755379051n_real @ A4 ) )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ord_less_real @ X @ X4 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_2014_Min__gr__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_rat @ X @ ( lattic8086005427650270231in_rat @ A4 ) )
          = ( ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_rat @ X @ X4 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_2015_Min__gr__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_num @ X @ ( lattic5278467273892544601in_num @ A4 ) )
          = ( ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_num @ X @ X4 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_2016_Min__gr__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A4 ) )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_int @ X @ X4 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_2017_Min__gr__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A4 ) )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_nat @ X @ X4 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_2018_Min_OcoboundedI,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A4 ) @ A ) ) ) ).

% Min.coboundedI
thf(fact_2019_Min_OcoboundedI,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A @ A4 )
       => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ A4 ) @ A ) ) ) ).

% Min.coboundedI
thf(fact_2020_Min_OcoboundedI,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ A4 ) @ A ) ) ) ).

% Min.coboundedI
thf(fact_2021_Min_OcoboundedI,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ( ord_less_eq_num @ ( lattic5278467273892544601in_num @ A4 ) @ A ) ) ) ).

% Min.coboundedI
thf(fact_2022_Min_OcoboundedI,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A4 ) @ A ) ) ) ).

% Min.coboundedI
thf(fact_2023_Min_OcoboundedI,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ A ) ) ) ).

% Min.coboundedI
thf(fact_2024_Min__eqI,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ A4 )
           => ( ord_less_eq_real @ X @ Y3 ) )
       => ( ( member_real @ X @ A4 )
         => ( ( lattic3629708407755379051n_real @ A4 )
            = X ) ) ) ) ).

% Min_eqI
thf(fact_2025_Min__eqI,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [Y3: code_integer] :
            ( ( member_Code_integer @ Y3 @ A4 )
           => ( ord_le3102999989581377725nteger @ X @ Y3 ) )
       => ( ( member_Code_integer @ X @ A4 )
         => ( ( lattic1063845414844153500nteger @ A4 )
            = X ) ) ) ) ).

% Min_eqI
thf(fact_2026_Min__eqI,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ! [Y3: rat] :
            ( ( member_rat @ Y3 @ A4 )
           => ( ord_less_eq_rat @ X @ Y3 ) )
       => ( ( member_rat @ X @ A4 )
         => ( ( lattic8086005427650270231in_rat @ A4 )
            = X ) ) ) ) ).

% Min_eqI
thf(fact_2027_Min__eqI,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ! [Y3: num] :
            ( ( member_num @ Y3 @ A4 )
           => ( ord_less_eq_num @ X @ Y3 ) )
       => ( ( member_num @ X @ A4 )
         => ( ( lattic5278467273892544601in_num @ A4 )
            = X ) ) ) ) ).

% Min_eqI
thf(fact_2028_Min__eqI,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ A4 )
           => ( ord_less_eq_int @ X @ Y3 ) )
       => ( ( member_int @ X @ A4 )
         => ( ( lattic8718645017227715691in_int @ A4 )
            = X ) ) ) ) ).

% Min_eqI
thf(fact_2029_Min__eqI,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A4 )
           => ( ord_less_eq_nat @ X @ Y3 ) )
       => ( ( member_nat @ X @ A4 )
         => ( ( lattic8721135487736765967in_nat @ A4 )
            = X ) ) ) ) ).

% Min_eqI
thf(fact_2030_Min__le,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A4 ) @ X ) ) ) ).

% Min_le
thf(fact_2031_Min__le,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ A4 ) @ X ) ) ) ).

% Min_le
thf(fact_2032_Min__le,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ X @ A4 )
       => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ A4 ) @ X ) ) ) ).

% Min_le
thf(fact_2033_Min__le,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ X @ A4 )
       => ( ord_less_eq_num @ ( lattic5278467273892544601in_num @ A4 ) @ X ) ) ) ).

% Min_le
thf(fact_2034_Min__le,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A4 ) @ X ) ) ) ).

% Min_le
thf(fact_2035_Min__le,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ X ) ) ) ).

% Min_le
thf(fact_2036_Min__in,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( member_real @ ( lattic3629708407755379051n_real @ A4 ) @ A4 ) ) ) ).

% Min_in
thf(fact_2037_Min__in,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( member_Code_integer @ ( lattic1063845414844153500nteger @ A4 ) @ A4 ) ) ) ).

% Min_in
thf(fact_2038_Min__in,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( member_int @ ( lattic8718645017227715691in_int @ A4 ) @ A4 ) ) ) ).

% Min_in
thf(fact_2039_Min__in,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( member_o @ ( lattic1973801136483472281_Min_o @ A4 ) @ A4 ) ) ) ).

% Min_in
thf(fact_2040_Min__in,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( member_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ A4 ) ) ) ).

% Min_in
thf(fact_2041_Max_OcoboundedI,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ( ord_less_eq_real @ A @ ( lattic4275903605611617917x_real @ A4 ) ) ) ) ).

% Max.coboundedI
thf(fact_2042_Max_OcoboundedI,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A @ A4 )
       => ( ord_le3102999989581377725nteger @ A @ ( lattic4901227151466704046nteger @ A4 ) ) ) ) ).

% Max.coboundedI
thf(fact_2043_Max_OcoboundedI,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ( ord_less_eq_rat @ A @ ( lattic7630753665789217321ax_rat @ A4 ) ) ) ) ).

% Max.coboundedI
thf(fact_2044_Max_OcoboundedI,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ( ord_less_eq_num @ A @ ( lattic4823215512031491691ax_num @ A4 ) ) ) ) ).

% Max.coboundedI
thf(fact_2045_Max_OcoboundedI,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ( ord_less_eq_int @ A @ ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ).

% Max.coboundedI
thf(fact_2046_Max_OcoboundedI,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).

% Max.coboundedI
thf(fact_2047_Max__eq__if,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ? [Xa: code_integer] :
                  ( ( member_Code_integer @ Xa @ B4 )
                  & ( ord_le3102999989581377725nteger @ X3 @ Xa ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ B4 )
               => ? [Xa: code_integer] :
                    ( ( member_Code_integer @ Xa @ A4 )
                    & ( ord_le3102999989581377725nteger @ X3 @ Xa ) ) )
           => ( ( lattic4901227151466704046nteger @ A4 )
              = ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_2048_Max__eq__if,axiom,
    ! [A4: set_rat,B4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( finite_finite_rat @ B4 )
       => ( ! [X3: rat] :
              ( ( member_rat @ X3 @ A4 )
             => ? [Xa: rat] :
                  ( ( member_rat @ Xa @ B4 )
                  & ( ord_less_eq_rat @ X3 @ Xa ) ) )
         => ( ! [X3: rat] :
                ( ( member_rat @ X3 @ B4 )
               => ? [Xa: rat] :
                    ( ( member_rat @ Xa @ A4 )
                    & ( ord_less_eq_rat @ X3 @ Xa ) ) )
           => ( ( lattic7630753665789217321ax_rat @ A4 )
              = ( lattic7630753665789217321ax_rat @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_2049_Max__eq__if,axiom,
    ! [A4: set_num,B4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( finite_finite_num @ B4 )
       => ( ! [X3: num] :
              ( ( member_num @ X3 @ A4 )
             => ? [Xa: num] :
                  ( ( member_num @ Xa @ B4 )
                  & ( ord_less_eq_num @ X3 @ Xa ) ) )
         => ( ! [X3: num] :
                ( ( member_num @ X3 @ B4 )
               => ? [Xa: num] :
                    ( ( member_num @ Xa @ A4 )
                    & ( ord_less_eq_num @ X3 @ Xa ) ) )
           => ( ( lattic4823215512031491691ax_num @ A4 )
              = ( lattic4823215512031491691ax_num @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_2050_Max__eq__if,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B4 )
                  & ( ord_less_eq_int @ X3 @ Xa ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ B4 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ A4 )
                    & ( ord_less_eq_int @ X3 @ Xa ) ) )
           => ( ( lattic8263393255366662781ax_int @ A4 )
              = ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_2051_Max__eq__if,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B4 )
                  & ( ord_less_eq_nat @ X3 @ Xa ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ B4 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ A4 )
                    & ( ord_less_eq_nat @ X3 @ Xa ) ) )
           => ( ( lattic8265883725875713057ax_nat @ A4 )
              = ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_2052_Max__eqI,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ A4 )
           => ( ord_less_eq_real @ Y3 @ X ) )
       => ( ( member_real @ X @ A4 )
         => ( ( lattic4275903605611617917x_real @ A4 )
            = X ) ) ) ) ).

% Max_eqI
thf(fact_2053_Max__eqI,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [Y3: code_integer] :
            ( ( member_Code_integer @ Y3 @ A4 )
           => ( ord_le3102999989581377725nteger @ Y3 @ X ) )
       => ( ( member_Code_integer @ X @ A4 )
         => ( ( lattic4901227151466704046nteger @ A4 )
            = X ) ) ) ) ).

% Max_eqI
thf(fact_2054_Max__eqI,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ! [Y3: rat] :
            ( ( member_rat @ Y3 @ A4 )
           => ( ord_less_eq_rat @ Y3 @ X ) )
       => ( ( member_rat @ X @ A4 )
         => ( ( lattic7630753665789217321ax_rat @ A4 )
            = X ) ) ) ) ).

% Max_eqI
thf(fact_2055_Max__eqI,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ! [Y3: num] :
            ( ( member_num @ Y3 @ A4 )
           => ( ord_less_eq_num @ Y3 @ X ) )
       => ( ( member_num @ X @ A4 )
         => ( ( lattic4823215512031491691ax_num @ A4 )
            = X ) ) ) ) ).

% Max_eqI
thf(fact_2056_Max__eqI,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ A4 )
           => ( ord_less_eq_int @ Y3 @ X ) )
       => ( ( member_int @ X @ A4 )
         => ( ( lattic8263393255366662781ax_int @ A4 )
            = X ) ) ) ) ).

% Max_eqI
thf(fact_2057_Max__eqI,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A4 )
           => ( ord_less_eq_nat @ Y3 @ X ) )
       => ( ( member_nat @ X @ A4 )
         => ( ( lattic8265883725875713057ax_nat @ A4 )
            = X ) ) ) ) ).

% Max_eqI
thf(fact_2058_Max__ge,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ord_less_eq_real @ X @ ( lattic4275903605611617917x_real @ A4 ) ) ) ) ).

% Max_ge
thf(fact_2059_Max__ge,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ord_le3102999989581377725nteger @ X @ ( lattic4901227151466704046nteger @ A4 ) ) ) ) ).

% Max_ge
thf(fact_2060_Max__ge,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ X @ A4 )
       => ( ord_less_eq_rat @ X @ ( lattic7630753665789217321ax_rat @ A4 ) ) ) ) ).

% Max_ge
thf(fact_2061_Max__ge,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ X @ A4 )
       => ( ord_less_eq_num @ X @ ( lattic4823215512031491691ax_num @ A4 ) ) ) ) ).

% Max_ge
thf(fact_2062_Max__ge,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ).

% Max_ge
thf(fact_2063_Max__ge,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).

% Max_ge
thf(fact_2064_Max__in,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( member_real @ ( lattic4275903605611617917x_real @ A4 ) @ A4 ) ) ) ).

% Max_in
thf(fact_2065_Max__in,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( member_Code_integer @ ( lattic4901227151466704046nteger @ A4 ) @ A4 ) ) ) ).

% Max_in
thf(fact_2066_Max__in,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( member_int @ ( lattic8263393255366662781ax_int @ A4 ) @ A4 ) ) ) ).

% Max_in
thf(fact_2067_Max__in,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( member_o @ ( lattic1921953407002678535_Max_o @ A4 ) @ A4 ) ) ) ).

% Max_in
thf(fact_2068_Max__in,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( member_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ A4 ) ) ) ).

% Max_in
thf(fact_2069_Min__eq__iff,axiom,
    ! [A4: set_real,M: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ( lattic3629708407755379051n_real @ A4 )
            = M )
          = ( ( member_real @ M @ A4 )
            & ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ord_less_eq_real @ M @ X4 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2070_Min__eq__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ( lattic1063845414844153500nteger @ A4 )
            = M )
          = ( ( member_Code_integer @ M @ A4 )
            & ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le3102999989581377725nteger @ M @ X4 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2071_Min__eq__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ( lattic1973801136483472281_Min_o @ A4 )
            = M )
          = ( ( member_o @ M @ A4 )
            & ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_eq_o @ M @ X4 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2072_Min__eq__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ( lattic8086005427650270231in_rat @ A4 )
            = M )
          = ( ( member_rat @ M @ A4 )
            & ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_eq_rat @ M @ X4 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2073_Min__eq__iff,axiom,
    ! [A4: set_num,M: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ( lattic5278467273892544601in_num @ A4 )
            = M )
          = ( ( member_num @ M @ A4 )
            & ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_eq_num @ M @ X4 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2074_Min__eq__iff,axiom,
    ! [A4: set_int,M: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ( lattic8718645017227715691in_int @ A4 )
            = M )
          = ( ( member_int @ M @ A4 )
            & ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_eq_int @ M @ X4 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2075_Min__eq__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ( lattic8721135487736765967in_nat @ A4 )
            = M )
          = ( ( member_nat @ M @ A4 )
            & ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_eq_nat @ M @ X4 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2076_Min__le__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ A4 ) @ X )
          = ( ? [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( ord_le3102999989581377725nteger @ X4 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2077_Min__le__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ A4 ) @ X )
          = ( ? [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( ord_less_eq_o @ X4 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2078_Min__le__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ A4 ) @ X )
          = ( ? [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
                & ( ord_less_eq_rat @ X4 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2079_Min__le__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ ( lattic5278467273892544601in_num @ A4 ) @ X )
          = ( ? [X4: num] :
                ( ( member_num @ X4 @ A4 )
                & ( ord_less_eq_num @ X4 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2080_Min__le__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A4 ) @ X )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( ord_less_eq_int @ X4 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2081_Min__le__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ X )
          = ( ? [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2082_eq__Min__iff,axiom,
    ! [A4: set_real,M: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( M
            = ( lattic3629708407755379051n_real @ A4 ) )
          = ( ( member_real @ M @ A4 )
            & ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ord_less_eq_real @ M @ X4 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2083_eq__Min__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( M
            = ( lattic1063845414844153500nteger @ A4 ) )
          = ( ( member_Code_integer @ M @ A4 )
            & ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le3102999989581377725nteger @ M @ X4 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2084_eq__Min__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( M
            = ( lattic1973801136483472281_Min_o @ A4 ) )
          = ( ( member_o @ M @ A4 )
            & ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_eq_o @ M @ X4 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2085_eq__Min__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( M
            = ( lattic8086005427650270231in_rat @ A4 ) )
          = ( ( member_rat @ M @ A4 )
            & ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_eq_rat @ M @ X4 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2086_eq__Min__iff,axiom,
    ! [A4: set_num,M: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( M
            = ( lattic5278467273892544601in_num @ A4 ) )
          = ( ( member_num @ M @ A4 )
            & ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_eq_num @ M @ X4 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2087_eq__Min__iff,axiom,
    ! [A4: set_int,M: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( M
            = ( lattic8718645017227715691in_int @ A4 ) )
          = ( ( member_int @ M @ A4 )
            & ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_eq_int @ M @ X4 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2088_eq__Min__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( M
            = ( lattic8721135487736765967in_nat @ A4 ) )
          = ( ( member_nat @ M @ A4 )
            & ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_eq_nat @ M @ X4 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2089_Min_OboundedE,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A4 ) )
         => ! [A8: real] :
              ( ( member_real @ A8 @ A4 )
             => ( ord_less_eq_real @ X @ A8 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2090_Min_OboundedE,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A4 ) )
         => ! [A8: code_integer] :
              ( ( member_Code_integer @ A8 @ A4 )
             => ( ord_le3102999989581377725nteger @ X @ A8 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2091_Min_OboundedE,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A4 ) )
         => ! [A8: $o] :
              ( ( member_o @ A8 @ A4 )
             => ( ord_less_eq_o @ X @ A8 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2092_Min_OboundedE,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A4 ) )
         => ! [A8: rat] :
              ( ( member_rat @ A8 @ A4 )
             => ( ord_less_eq_rat @ X @ A8 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2093_Min_OboundedE,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ X @ ( lattic5278467273892544601in_num @ A4 ) )
         => ! [A8: num] :
              ( ( member_num @ A8 @ A4 )
             => ( ord_less_eq_num @ X @ A8 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2094_Min_OboundedE,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A4 ) )
         => ! [A8: int] :
              ( ( member_int @ A8 @ A4 )
             => ( ord_less_eq_int @ X @ A8 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2095_Min_OboundedE,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A4 ) )
         => ! [A8: nat] :
              ( ( member_nat @ A8 @ A4 )
             => ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2096_Min_OboundedI,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( ord_less_eq_real @ X @ A3 ) )
         => ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A4 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2097_Min_OboundedI,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A4 )
             => ( ord_le3102999989581377725nteger @ X @ A3 ) )
         => ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A4 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2098_Min_OboundedI,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ A4 )
             => ( ord_less_eq_o @ X @ A3 ) )
         => ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A4 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2099_Min_OboundedI,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( ( member_rat @ A3 @ A4 )
             => ( ord_less_eq_rat @ X @ A3 ) )
         => ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A4 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2100_Min_OboundedI,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ! [A3: num] :
              ( ( member_num @ A3 @ A4 )
             => ( ord_less_eq_num @ X @ A3 ) )
         => ( ord_less_eq_num @ X @ ( lattic5278467273892544601in_num @ A4 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2101_Min_OboundedI,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( ord_less_eq_int @ X @ A3 ) )
         => ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A4 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2102_Min_OboundedI,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
             => ( ord_less_eq_nat @ X @ A3 ) )
         => ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2103_Min__less__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le6747313008572928689nteger @ ( lattic1063845414844153500nteger @ A4 ) @ X )
          = ( ? [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( ord_le6747313008572928689nteger @ X4 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_2104_Min__less__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_o @ ( lattic1973801136483472281_Min_o @ A4 ) @ X )
          = ( ? [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( ord_less_o @ X4 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_2105_Min__less__iff,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ord_less_real @ ( lattic3629708407755379051n_real @ A4 ) @ X )
          = ( ? [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( ord_less_real @ X4 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_2106_Min__less__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_rat @ ( lattic8086005427650270231in_rat @ A4 ) @ X )
          = ( ? [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
                & ( ord_less_rat @ X4 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_2107_Min__less__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_num @ ( lattic5278467273892544601in_num @ A4 ) @ X )
          = ( ? [X4: num] :
                ( ( member_num @ X4 @ A4 )
                & ( ord_less_num @ X4 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_2108_Min__less__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A4 ) @ X )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( ord_less_int @ X4 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_2109_Min__less__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ X )
          = ( ? [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( ord_less_nat @ X4 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_2110_Min__insert2,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ! [B3: $o] :
            ( ( member_o @ B3 @ A4 )
           => ( ord_less_eq_o @ A @ B3 ) )
       => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ A @ A4 ) )
          = A ) ) ) ).

% Min_insert2
thf(fact_2111_Min__insert2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [B3: real] :
            ( ( member_real @ B3 @ A4 )
           => ( ord_less_eq_real @ A @ B3 ) )
       => ( ( lattic3629708407755379051n_real @ ( insert_real @ A @ A4 ) )
          = A ) ) ) ).

% Min_insert2
thf(fact_2112_Min__insert2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [B3: code_integer] :
            ( ( member_Code_integer @ B3 @ A4 )
           => ( ord_le3102999989581377725nteger @ A @ B3 ) )
       => ( ( lattic1063845414844153500nteger @ ( insert_Code_integer @ A @ A4 ) )
          = A ) ) ) ).

% Min_insert2
thf(fact_2113_Min__insert2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ! [B3: rat] :
            ( ( member_rat @ B3 @ A4 )
           => ( ord_less_eq_rat @ A @ B3 ) )
       => ( ( lattic8086005427650270231in_rat @ ( insert_rat @ A @ A4 ) )
          = A ) ) ) ).

% Min_insert2
thf(fact_2114_Min__insert2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ! [B3: num] :
            ( ( member_num @ B3 @ A4 )
           => ( ord_less_eq_num @ A @ B3 ) )
       => ( ( lattic5278467273892544601in_num @ ( insert_num @ A @ A4 ) )
          = A ) ) ) ).

% Min_insert2
thf(fact_2115_Min__insert2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ! [B3: int] :
            ( ( member_int @ B3 @ A4 )
           => ( ord_less_eq_int @ A @ B3 ) )
       => ( ( lattic8718645017227715691in_int @ ( insert_int @ A @ A4 ) )
          = A ) ) ) ).

% Min_insert2
thf(fact_2116_Min__insert2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [B3: nat] :
            ( ( member_nat @ B3 @ A4 )
           => ( ord_less_eq_nat @ A @ B3 ) )
       => ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A @ A4 ) )
          = A ) ) ) ).

% Min_insert2
thf(fact_2117_Max__eq__iff,axiom,
    ! [A4: set_real,M: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ( lattic4275903605611617917x_real @ A4 )
            = M )
          = ( ( member_real @ M @ A4 )
            & ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ord_less_eq_real @ X4 @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2118_Max__eq__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ( lattic4901227151466704046nteger @ A4 )
            = M )
          = ( ( member_Code_integer @ M @ A4 )
            & ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le3102999989581377725nteger @ X4 @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2119_Max__eq__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ( lattic1921953407002678535_Max_o @ A4 )
            = M )
          = ( ( member_o @ M @ A4 )
            & ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_eq_o @ X4 @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2120_Max__eq__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ( lattic7630753665789217321ax_rat @ A4 )
            = M )
          = ( ( member_rat @ M @ A4 )
            & ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_eq_rat @ X4 @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2121_Max__eq__iff,axiom,
    ! [A4: set_num,M: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ( lattic4823215512031491691ax_num @ A4 )
            = M )
          = ( ( member_num @ M @ A4 )
            & ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_eq_num @ X4 @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2122_Max__eq__iff,axiom,
    ! [A4: set_int,M: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ( lattic8263393255366662781ax_int @ A4 )
            = M )
          = ( ( member_int @ M @ A4 )
            & ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_eq_int @ X4 @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2123_Max__eq__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ( lattic8265883725875713057ax_nat @ A4 )
            = M )
          = ( ( member_nat @ M @ A4 )
            & ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_eq_nat @ X4 @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2124_Max__ge__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ X @ ( lattic4901227151466704046nteger @ A4 ) )
          = ( ? [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( ord_le3102999989581377725nteger @ X @ X4 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2125_Max__ge__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic1921953407002678535_Max_o @ A4 ) )
          = ( ? [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( ord_less_eq_o @ X @ X4 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2126_Max__ge__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ X @ ( lattic7630753665789217321ax_rat @ A4 ) )
          = ( ? [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
                & ( ord_less_eq_rat @ X @ X4 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2127_Max__ge__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ X @ ( lattic4823215512031491691ax_num @ A4 ) )
          = ( ? [X4: num] :
                ( ( member_num @ X4 @ A4 )
                & ( ord_less_eq_num @ X @ X4 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2128_Max__ge__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( ord_less_eq_int @ X @ X4 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2129_Max__ge__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) )
          = ( ? [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2130_eq__Max__iff,axiom,
    ! [A4: set_real,M: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( M
            = ( lattic4275903605611617917x_real @ A4 ) )
          = ( ( member_real @ M @ A4 )
            & ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ord_less_eq_real @ X4 @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2131_eq__Max__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( M
            = ( lattic4901227151466704046nteger @ A4 ) )
          = ( ( member_Code_integer @ M @ A4 )
            & ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ord_le3102999989581377725nteger @ X4 @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2132_eq__Max__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( M
            = ( lattic1921953407002678535_Max_o @ A4 ) )
          = ( ( member_o @ M @ A4 )
            & ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ord_less_eq_o @ X4 @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2133_eq__Max__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( M
            = ( lattic7630753665789217321ax_rat @ A4 ) )
          = ( ( member_rat @ M @ A4 )
            & ! [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
               => ( ord_less_eq_rat @ X4 @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2134_eq__Max__iff,axiom,
    ! [A4: set_num,M: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( M
            = ( lattic4823215512031491691ax_num @ A4 ) )
          = ( ( member_num @ M @ A4 )
            & ! [X4: num] :
                ( ( member_num @ X4 @ A4 )
               => ( ord_less_eq_num @ X4 @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2135_eq__Max__iff,axiom,
    ! [A4: set_int,M: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( M
            = ( lattic8263393255366662781ax_int @ A4 ) )
          = ( ( member_int @ M @ A4 )
            & ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ord_less_eq_int @ X4 @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2136_eq__Max__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( M
            = ( lattic8265883725875713057ax_nat @ A4 ) )
          = ( ( member_nat @ M @ A4 )
            & ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ord_less_eq_nat @ X4 @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2137_Max_OboundedE,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A4 ) @ X )
         => ! [A8: real] :
              ( ( member_real @ A8 @ A4 )
             => ( ord_less_eq_real @ A8 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2138_Max_OboundedE,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A4 ) @ X )
         => ! [A8: code_integer] :
              ( ( member_Code_integer @ A8 @ A4 )
             => ( ord_le3102999989581377725nteger @ A8 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2139_Max_OboundedE,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A4 ) @ X )
         => ! [A8: $o] :
              ( ( member_o @ A8 @ A4 )
             => ( ord_less_eq_o @ A8 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2140_Max_OboundedE,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A4 ) @ X )
         => ! [A8: rat] :
              ( ( member_rat @ A8 @ A4 )
             => ( ord_less_eq_rat @ A8 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2141_Max_OboundedE,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A4 ) @ X )
         => ! [A8: num] :
              ( ( member_num @ A8 @ A4 )
             => ( ord_less_eq_num @ A8 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2142_Max_OboundedE,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X )
         => ! [A8: int] :
              ( ( member_int @ A8 @ A4 )
             => ( ord_less_eq_int @ A8 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2143_Max_OboundedE,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X )
         => ! [A8: nat] :
              ( ( member_nat @ A8 @ A4 )
             => ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2144_Max_OboundedI,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( ord_less_eq_real @ A3 @ X ) )
         => ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A4 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2145_Max_OboundedI,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A4 )
             => ( ord_le3102999989581377725nteger @ A3 @ X ) )
         => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A4 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2146_Max_OboundedI,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ A4 )
             => ( ord_less_eq_o @ A3 @ X ) )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A4 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2147_Max_OboundedI,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( ( member_rat @ A3 @ A4 )
             => ( ord_less_eq_rat @ A3 @ X ) )
         => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A4 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2148_Max_OboundedI,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ! [A3: num] :
              ( ( member_num @ A3 @ A4 )
             => ( ord_less_eq_num @ A3 @ X ) )
         => ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A4 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2149_Max_OboundedI,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( ord_less_eq_int @ A3 @ X ) )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2150_Max_OboundedI,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
             => ( ord_less_eq_nat @ A3 @ X ) )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2151_Max__gr__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le6747313008572928689nteger @ X @ ( lattic4901227151466704046nteger @ A4 ) )
          = ( ? [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( ord_le6747313008572928689nteger @ X @ X4 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_2152_Max__gr__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( ord_less_o @ X @ ( lattic1921953407002678535_Max_o @ A4 ) )
          = ( ? [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( ord_less_o @ X @ X4 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_2153_Max__gr__iff,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ( ord_less_real @ X @ ( lattic4275903605611617917x_real @ A4 ) )
          = ( ? [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( ord_less_real @ X @ X4 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_2154_Max__gr__iff,axiom,
    ! [A4: set_rat,X: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( ord_less_rat @ X @ ( lattic7630753665789217321ax_rat @ A4 ) )
          = ( ? [X4: rat] :
                ( ( member_rat @ X4 @ A4 )
                & ( ord_less_rat @ X @ X4 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_2155_Max__gr__iff,axiom,
    ! [A4: set_num,X: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( ord_less_num @ X @ ( lattic4823215512031491691ax_num @ A4 ) )
          = ( ? [X4: num] :
                ( ( member_num @ X4 @ A4 )
                & ( ord_less_num @ X @ X4 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_2156_Max__gr__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( ord_less_int @ X @ X4 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_2157_Max__gr__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) )
          = ( ? [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( ord_less_nat @ X @ X4 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_2158_Max__insert2,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ! [B3: $o] :
            ( ( member_o @ B3 @ A4 )
           => ( ord_less_eq_o @ B3 @ A ) )
       => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ A @ A4 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_2159_Max__insert2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [B3: real] :
            ( ( member_real @ B3 @ A4 )
           => ( ord_less_eq_real @ B3 @ A ) )
       => ( ( lattic4275903605611617917x_real @ ( insert_real @ A @ A4 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_2160_Max__insert2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [B3: code_integer] :
            ( ( member_Code_integer @ B3 @ A4 )
           => ( ord_le3102999989581377725nteger @ B3 @ A ) )
       => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ A @ A4 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_2161_Max__insert2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ! [B3: rat] :
            ( ( member_rat @ B3 @ A4 )
           => ( ord_less_eq_rat @ B3 @ A ) )
       => ( ( lattic7630753665789217321ax_rat @ ( insert_rat @ A @ A4 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_2162_Max__insert2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ! [B3: num] :
            ( ( member_num @ B3 @ A4 )
           => ( ord_less_eq_num @ B3 @ A ) )
       => ( ( lattic4823215512031491691ax_num @ ( insert_num @ A @ A4 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_2163_Max__insert2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ! [B3: int] :
            ( ( member_int @ B3 @ A4 )
           => ( ord_less_eq_int @ B3 @ A ) )
       => ( ( lattic8263393255366662781ax_int @ ( insert_int @ A @ A4 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_2164_Max__insert2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [B3: nat] :
            ( ( member_nat @ B3 @ A4 )
           => ( ord_less_eq_nat @ B3 @ A ) )
       => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A @ A4 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_2165_Min__antimono,axiom,
    ! [M7: set_Code_integer,N8: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ M7 @ N8 )
     => ( ( M7 != bot_bo3990330152332043303nteger )
       => ( ( finite6017078050557962740nteger @ N8 )
         => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ N8 ) @ ( lattic1063845414844153500nteger @ M7 ) ) ) ) ) ).

% Min_antimono
thf(fact_2166_Min__antimono,axiom,
    ! [M7: set_o,N8: set_o] :
      ( ( ord_less_eq_set_o @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_o )
       => ( ( finite_finite_o @ N8 )
         => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ N8 ) @ ( lattic1973801136483472281_Min_o @ M7 ) ) ) ) ) ).

% Min_antimono
thf(fact_2167_Min__antimono,axiom,
    ! [M7: set_rat,N8: set_rat] :
      ( ( ord_less_eq_set_rat @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_rat )
       => ( ( finite_finite_rat @ N8 )
         => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ N8 ) @ ( lattic8086005427650270231in_rat @ M7 ) ) ) ) ) ).

% Min_antimono
thf(fact_2168_Min__antimono,axiom,
    ! [M7: set_num,N8: set_num] :
      ( ( ord_less_eq_set_num @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_num )
       => ( ( finite_finite_num @ N8 )
         => ( ord_less_eq_num @ ( lattic5278467273892544601in_num @ N8 ) @ ( lattic5278467273892544601in_num @ M7 ) ) ) ) ) ).

% Min_antimono
thf(fact_2169_Min__antimono,axiom,
    ! [M7: set_int,N8: set_int] :
      ( ( ord_less_eq_set_int @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_int )
       => ( ( finite_finite_int @ N8 )
         => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N8 ) @ ( lattic8718645017227715691in_int @ M7 ) ) ) ) ) ).

% Min_antimono
thf(fact_2170_Min__antimono,axiom,
    ! [M7: set_nat,N8: set_nat] :
      ( ( ord_less_eq_set_nat @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N8 )
         => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N8 ) @ ( lattic8721135487736765967in_nat @ M7 ) ) ) ) ) ).

% Min_antimono
thf(fact_2171_Min_Osubset__imp,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( finite6017078050557962740nteger @ B4 )
         => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ B4 ) @ ( lattic1063845414844153500nteger @ A4 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2172_Min_Osubset__imp,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( finite_finite_o @ B4 )
         => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ B4 ) @ ( lattic1973801136483472281_Min_o @ A4 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2173_Min_Osubset__imp,axiom,
    ! [A4: set_rat,B4: set_rat] :
      ( ( ord_less_eq_set_rat @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( finite_finite_rat @ B4 )
         => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ B4 ) @ ( lattic8086005427650270231in_rat @ A4 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2174_Min_Osubset__imp,axiom,
    ! [A4: set_num,B4: set_num] :
      ( ( ord_less_eq_set_num @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( finite_finite_num @ B4 )
         => ( ord_less_eq_num @ ( lattic5278467273892544601in_num @ B4 ) @ ( lattic5278467273892544601in_num @ A4 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2175_Min_Osubset__imp,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( finite_finite_int @ B4 )
         => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B4 ) @ ( lattic8718645017227715691in_int @ A4 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2176_Min_Osubset__imp,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B4 )
         => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B4 ) @ ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2177_Max__mono,axiom,
    ! [M7: set_Code_integer,N8: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ M7 @ N8 )
     => ( ( M7 != bot_bo3990330152332043303nteger )
       => ( ( finite6017078050557962740nteger @ N8 )
         => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ M7 ) @ ( lattic4901227151466704046nteger @ N8 ) ) ) ) ) ).

% Max_mono
thf(fact_2178_Max__mono,axiom,
    ! [M7: set_o,N8: set_o] :
      ( ( ord_less_eq_set_o @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_o )
       => ( ( finite_finite_o @ N8 )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ M7 ) @ ( lattic1921953407002678535_Max_o @ N8 ) ) ) ) ) ).

% Max_mono
thf(fact_2179_Max__mono,axiom,
    ! [M7: set_rat,N8: set_rat] :
      ( ( ord_less_eq_set_rat @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_rat )
       => ( ( finite_finite_rat @ N8 )
         => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ M7 ) @ ( lattic7630753665789217321ax_rat @ N8 ) ) ) ) ) ).

% Max_mono
thf(fact_2180_Max__mono,axiom,
    ! [M7: set_num,N8: set_num] :
      ( ( ord_less_eq_set_num @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_num )
       => ( ( finite_finite_num @ N8 )
         => ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ M7 ) @ ( lattic4823215512031491691ax_num @ N8 ) ) ) ) ) ).

% Max_mono
thf(fact_2181_Max__mono,axiom,
    ! [M7: set_int,N8: set_int] :
      ( ( ord_less_eq_set_int @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_int )
       => ( ( finite_finite_int @ N8 )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M7 ) @ ( lattic8263393255366662781ax_int @ N8 ) ) ) ) ) ).

% Max_mono
thf(fact_2182_Max__mono,axiom,
    ! [M7: set_nat,N8: set_nat] :
      ( ( ord_less_eq_set_nat @ M7 @ N8 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N8 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M7 ) @ ( lattic8265883725875713057ax_nat @ N8 ) ) ) ) ) ).

% Max_mono
thf(fact_2183_Max_Osubset__imp,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( finite6017078050557962740nteger @ B4 )
         => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A4 ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2184_Max_Osubset__imp,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( finite_finite_o @ B4 )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A4 ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2185_Max_Osubset__imp,axiom,
    ! [A4: set_rat,B4: set_rat] :
      ( ( ord_less_eq_set_rat @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_rat )
       => ( ( finite_finite_rat @ B4 )
         => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A4 ) @ ( lattic7630753665789217321ax_rat @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2186_Max_Osubset__imp,axiom,
    ! [A4: set_num,B4: set_num] :
      ( ( ord_less_eq_set_num @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_num )
       => ( ( finite_finite_num @ B4 )
         => ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A4 ) @ ( lattic4823215512031491691ax_num @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2187_Max_Osubset__imp,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( finite_finite_int @ B4 )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2188_Max_Osubset__imp,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B4 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2189_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y4: nat,X4: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ).

% greater_shift
thf(fact_2190_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] : ( vEBT_VEBT_less @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ).

% less_shift
thf(fact_2191_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,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_2192_the__elem__eq,axiom,
    ! [X: vEBT_VEBT] :
      ( ( the_elem_VEBT_VEBT @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
      = X ) ).

% the_elem_eq
thf(fact_2193_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_2194_the__elem__eq,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% the_elem_eq
thf(fact_2195_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_2196_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_2197_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_2198_is__singletonI,axiom,
    ! [X: vEBT_VEBT] : ( is_sin24926331636114728T_VEBT @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% is_singletonI
thf(fact_2199_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_2200_is__singletonI,axiom,
    ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_2201_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_2202_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_2203_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_2204_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_2205_is__singleton__the__elem,axiom,
    ( is_sin24926331636114728T_VEBT
    = ( ^ [A6: set_VEBT_VEBT] :
          ( A6
          = ( insert_VEBT_VEBT @ ( the_elem_VEBT_VEBT @ A6 ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2206_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( A6
          = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2207_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
          ( A6
          = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2208_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
          ( A6
          = ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2209_is__singletonI_H,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( A4 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
           => ( ( member_VEBT_VEBT @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_sin24926331636114728T_VEBT @ A4 ) ) ) ).

% is_singletonI'
thf(fact_2210_is__singletonI_H,axiom,
    ! [A4: set_real] :
      ( ( A4 != bot_bot_set_real )
     => ( ! [X3: real,Y3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ( member_real @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_real @ A4 ) ) ) ).

% is_singletonI'
thf(fact_2211_is__singletonI_H,axiom,
    ! [A4: set_set_nat] :
      ( ( A4 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat,Y3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
           => ( ( member_set_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_set_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_2212_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_2213_is__singletonI_H,axiom,
    ! [A4: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X3: int,Y3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ( member_int @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_int @ A4 ) ) ) ).

% is_singletonI'
thf(fact_2214_is__singletonI_H,axiom,
    ! [A4: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X3: $o,Y3: $o] :
            ( ( member_o @ X3 @ A4 )
           => ( ( member_o @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_o @ A4 ) ) ) ).

% is_singletonI'
thf(fact_2215_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_2216_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_2217_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,
    ! [Uv2: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_2218_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu2: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu2 @ $true ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_2219_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A @ B ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_2220_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_2221_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_2222_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,Va2: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_2223_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_2224_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz @ Va2 ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_2225_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_2226_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz @ Va2 ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_2227_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz ) @ Va2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_2228_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_2229_is__singletonE,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( is_sin24926331636114728T_VEBT @ A4 )
     => ~ ! [X3: vEBT_VEBT] :
            ( A4
           != ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% is_singletonE
thf(fact_2230_is__singletonE,axiom,
    ! [A4: set_nat] :
      ( ( is_singleton_nat @ A4 )
     => ~ ! [X3: nat] :
            ( A4
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_2231_is__singletonE,axiom,
    ! [A4: set_int] :
      ( ( is_singleton_int @ A4 )
     => ~ ! [X3: int] :
            ( A4
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_2232_is__singletonE,axiom,
    ! [A4: set_o] :
      ( ( is_singleton_o @ A4 )
     => ~ ! [X3: $o] :
            ( A4
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_2233_is__singleton__def,axiom,
    ( is_sin24926331636114728T_VEBT
    = ( ^ [A6: set_VEBT_VEBT] :
        ? [X4: vEBT_VEBT] :
          ( A6
          = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% is_singleton_def
thf(fact_2234_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
        ? [X4: nat] :
          ( A6
          = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_2235_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
        ? [X4: int] :
          ( A6
          = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_2236_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
        ? [X4: $o] :
          ( A6
          = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_2237_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz ) @ Va2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_2238_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,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_2239_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,
    ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_2240_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_2241_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_2242_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_2243_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd2 ) @ Ve2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_2244_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_2245_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd2 ) @ Ve2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_2246_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,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_2247_set__removeAll,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( removeAll_VEBT_VEBT @ X @ Xs ) )
      = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_removeAll
thf(fact_2248_set__removeAll,axiom,
    ! [X: real,Xs: list_real] :
      ( ( set_real2 @ ( removeAll_real @ X @ Xs ) )
      = ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_removeAll
thf(fact_2249_set__removeAll,axiom,
    ! [X: int,Xs: list_int] :
      ( ( set_int2 @ ( removeAll_int @ X @ Xs ) )
      = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_removeAll
thf(fact_2250_set__removeAll,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( set_o2 @ ( removeAll_o @ X @ Xs ) )
      = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% set_removeAll
thf(fact_2251_set__removeAll,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
      = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_removeAll
thf(fact_2252_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_2253_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_2254_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( 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_2255_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L ) @ one_one_nat ) ) ) ).

% insertsimp'
thf(fact_2256_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y ) @ one_one_nat ) ) ) ).

% insersimp'
thf(fact_2257_frac__unique__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = A )
      = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
        & ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ord_less_real @ A @ one_one_real ) ) ) ).

% frac_unique_iff
thf(fact_2258_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = A )
      = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ord_less_rat @ A @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_2259_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,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_2260_remove__def,axiom,
    ( remove_VEBT_VEBT
    = ( ^ [X4: vEBT_VEBT,A6: set_VEBT_VEBT] : ( minus_5127226145743854075T_VEBT @ A6 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% remove_def
thf(fact_2261_remove__def,axiom,
    ( remove_int
    = ( ^ [X4: int,A6: set_int] : ( minus_minus_set_int @ A6 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% remove_def
thf(fact_2262_remove__def,axiom,
    ( remove_o
    = ( ^ [X4: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_2263_remove__def,axiom,
    ( remove_nat
    = ( ^ [X4: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% remove_def
thf(fact_2264_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,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_2265_member__remove,axiom,
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( member_nat @ X @ ( remove_nat @ Y @ A4 ) )
      = ( ( member_nat @ X @ A4 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_2266_member__remove,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( remove_VEBT_VEBT @ Y @ A4 ) )
      = ( ( member_VEBT_VEBT @ X @ A4 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_2267_member__remove,axiom,
    ! [X: real,Y: real,A4: set_real] :
      ( ( member_real @ X @ ( remove_real @ Y @ A4 ) )
      = ( ( member_real @ X @ A4 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_2268_member__remove,axiom,
    ! [X: int,Y: int,A4: set_int] :
      ( ( member_int @ X @ ( remove_int @ Y @ A4 ) )
      = ( ( member_int @ X @ A4 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_2269_member__remove,axiom,
    ! [X: set_nat,Y: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ X @ ( remove_set_nat @ Y @ A4 ) )
      = ( ( member_set_nat @ X @ A4 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_2270_frac__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% frac_eq_0_iff
thf(fact_2271_frac__eq__0__iff,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = zero_zero_rat )
      = ( member_rat @ X @ ring_1_Ints_rat ) ) ).

% frac_eq_0_iff
thf(fact_2272_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
      = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).

% frac_gt_0_iff
thf(fact_2273_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
      = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).

% frac_gt_0_iff
thf(fact_2274_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_2275_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A @ B ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_2276_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,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_2277_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y = one_one_nat )
             => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu @ $true ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_2278_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% delete_bound_height'
thf(fact_2279_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_2280_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_2281_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_2282_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_2283_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_2284_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
         => ( ! [Uu: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_2285_Ints__0,axiom,
    member_uint32 @ zero_zero_uint32 @ ring_1_Ints_uint32 ).

% Ints_0
thf(fact_2286_Ints__0,axiom,
    member_real @ zero_zero_real @ ring_1_Ints_real ).

% Ints_0
thf(fact_2287_Ints__0,axiom,
    member_rat @ zero_zero_rat @ ring_1_Ints_rat ).

% Ints_0
thf(fact_2288_Ints__0,axiom,
    member_int @ zero_zero_int @ ring_1_Ints_int ).

% Ints_0
thf(fact_2289_enumerate__Suc_H,axiom,
    ! [S3: set_nat,N: nat] :
      ( ( infini8530281810654367211te_nat @ S3 @ ( suc @ N ) )
      = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ ( infini8530281810654367211te_nat @ S3 @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).

% enumerate_Suc'
thf(fact_2290_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] :
          ( X
         != ( plus_plus_nat @ N2 @ N2 ) )
     => ~ ! [N2: nat] :
            ( X
           != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).

% even_odd_cases
thf(fact_2291_add__right__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_2292_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_2293_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_2294_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_2295_add__left__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_2296_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_2297_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_2298_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_2299_image__eqI,axiom,
    ! [B: nat,F: nat > nat,X: nat,A4: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A4 )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2300_image__eqI,axiom,
    ! [B: vEBT_VEBT,F: nat > vEBT_VEBT,X: nat,A4: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A4 )
       => ( member_VEBT_VEBT @ B @ ( image_nat_VEBT_VEBT @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2301_image__eqI,axiom,
    ! [B: real,F: nat > real,X: nat,A4: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A4 )
       => ( member_real @ B @ ( image_nat_real @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2302_image__eqI,axiom,
    ! [B: int,F: nat > int,X: nat,A4: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A4 )
       => ( member_int @ B @ ( image_nat_int @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2303_image__eqI,axiom,
    ! [B: nat,F: vEBT_VEBT > nat,X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( member_nat @ B @ ( image_VEBT_VEBT_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2304_image__eqI,axiom,
    ! [B: vEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT,X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( member_VEBT_VEBT @ B @ ( image_3375948659692109573T_VEBT @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2305_image__eqI,axiom,
    ! [B: real,F: vEBT_VEBT > real,X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( member_real @ B @ ( image_VEBT_VEBT_real @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2306_image__eqI,axiom,
    ! [B: int,F: vEBT_VEBT > int,X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( member_int @ B @ ( image_VEBT_VEBT_int @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2307_image__eqI,axiom,
    ! [B: nat,F: real > nat,X: real,A4: set_real] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_real @ X @ A4 )
       => ( member_nat @ B @ ( image_real_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2308_image__eqI,axiom,
    ! [B: vEBT_VEBT,F: real > vEBT_VEBT,X: real,A4: set_real] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_real @ X @ A4 )
       => ( member_VEBT_VEBT @ B @ ( image_real_VEBT_VEBT @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_2309_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_2310_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_2311_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_2312_add_Oright__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% add.right_neutral
thf(fact_2313_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_2314_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_2315_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_2316_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_2317_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_2318_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2319_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2320_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_2321_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_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_2341_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_2342_add__0,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% add_0
thf(fact_2343_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_2344_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_2345_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_2346_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_2347_add__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2348_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2349_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2350_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2351_add__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2352_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2353_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2354_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2355_add__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2356_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2357_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2358_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2359_add__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2360_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2361_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2362_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2363_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_2364_add__diff__cancel__right_H,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ ( plus_plus_uint32 @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_2365_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_2366_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_2367_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_2368_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2369_add__diff__cancel__right,axiom,
    ! [A: uint32,C2: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ ( plus_plus_uint32 @ A @ C2 ) @ ( plus_plus_uint32 @ B @ C2 ) )
      = ( minus_minus_uint32 @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2370_add__diff__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2371_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2372_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2373_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_2374_add__diff__cancel__left_H,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ ( plus_plus_uint32 @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_2375_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_2376_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_2377_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_2378_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2379_add__diff__cancel__left,axiom,
    ! [C2: uint32,A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ ( plus_plus_uint32 @ C2 @ A ) @ ( plus_plus_uint32 @ C2 @ B ) )
      = ( minus_minus_uint32 @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2380_add__diff__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2381_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2382_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2383_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2384_diff__add__cancel,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( minus_minus_uint32 @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2385_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2386_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2387_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2388_add__diff__cancel,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ ( plus_plus_uint32 @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2389_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2390_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2391_image__empty,axiom,
    ! [F: nat > set_nat] :
      ( ( image_nat_set_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_set_nat ) ).

% image_empty
thf(fact_2392_image__empty,axiom,
    ! [F: nat > nat] :
      ( ( image_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_2393_image__empty,axiom,
    ! [F: nat > int] :
      ( ( image_nat_int @ F @ bot_bot_set_nat )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_2394_image__empty,axiom,
    ! [F: nat > $o] :
      ( ( image_nat_o @ F @ bot_bot_set_nat )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_2395_image__empty,axiom,
    ! [F: int > nat] :
      ( ( image_int_nat @ F @ bot_bot_set_int )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_2396_image__empty,axiom,
    ! [F: int > int] :
      ( ( image_int_int @ F @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_2397_image__empty,axiom,
    ! [F: int > $o] :
      ( ( image_int_o @ F @ bot_bot_set_int )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_2398_image__empty,axiom,
    ! [F: $o > nat] :
      ( ( image_o_nat @ F @ bot_bot_set_o )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_2399_image__empty,axiom,
    ! [F: $o > int] :
      ( ( image_o_int @ F @ bot_bot_set_o )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_2400_image__empty,axiom,
    ! [F: $o > $o] :
      ( ( image_o_o @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_2401_empty__is__image,axiom,
    ! [F: nat > set_nat,A4: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( image_nat_set_nat @ F @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_2402_empty__is__image,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_2403_empty__is__image,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( bot_bot_set_nat
        = ( image_int_nat @ F @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_2404_empty__is__image,axiom,
    ! [F: $o > nat,A4: set_o] :
      ( ( bot_bot_set_nat
        = ( image_o_nat @ F @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_2405_empty__is__image,axiom,
    ! [F: nat > int,A4: set_nat] :
      ( ( bot_bot_set_int
        = ( image_nat_int @ F @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_2406_empty__is__image,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( bot_bot_set_int
        = ( image_int_int @ F @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_2407_empty__is__image,axiom,
    ! [F: $o > int,A4: set_o] :
      ( ( bot_bot_set_int
        = ( image_o_int @ F @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_2408_empty__is__image,axiom,
    ! [F: nat > $o,A4: set_nat] :
      ( ( bot_bot_set_o
        = ( image_nat_o @ F @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_2409_empty__is__image,axiom,
    ! [F: int > $o,A4: set_int] :
      ( ( bot_bot_set_o
        = ( image_int_o @ F @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_2410_empty__is__image,axiom,
    ! [F: $o > $o,A4: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o @ F @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_2411_image__is__empty,axiom,
    ! [F: nat > set_nat,A4: set_nat] :
      ( ( ( image_nat_set_nat @ F @ A4 )
        = bot_bot_set_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_2412_image__is__empty,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( ( image_nat_nat @ F @ A4 )
        = bot_bot_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_2413_image__is__empty,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( ( image_int_nat @ F @ A4 )
        = bot_bot_set_nat )
      = ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_2414_image__is__empty,axiom,
    ! [F: $o > nat,A4: set_o] :
      ( ( ( image_o_nat @ F @ A4 )
        = bot_bot_set_nat )
      = ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_2415_image__is__empty,axiom,
    ! [F: nat > int,A4: set_nat] :
      ( ( ( image_nat_int @ F @ A4 )
        = bot_bot_set_int )
      = ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_2416_image__is__empty,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ( image_int_int @ F @ A4 )
        = bot_bot_set_int )
      = ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_2417_image__is__empty,axiom,
    ! [F: $o > int,A4: set_o] :
      ( ( ( image_o_int @ F @ A4 )
        = bot_bot_set_int )
      = ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_2418_image__is__empty,axiom,
    ! [F: nat > $o,A4: set_nat] :
      ( ( ( image_nat_o @ F @ A4 )
        = bot_bot_set_o )
      = ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_2419_image__is__empty,axiom,
    ! [F: int > $o,A4: set_int] :
      ( ( ( image_int_o @ F @ A4 )
        = bot_bot_set_o )
      = ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_2420_image__is__empty,axiom,
    ! [F: $o > $o,A4: set_o] :
      ( ( ( image_o_o @ F @ A4 )
        = bot_bot_set_o )
      = ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_2421_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_2422_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_nat @ ( image_nat_nat @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2423_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > int] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_int @ ( image_nat_int @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2424_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > complex] :
      ( ( finite_finite_nat @ F2 )
     => ( finite3207457112153483333omplex @ ( image_nat_complex @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2425_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > code_integer] :
      ( ( finite_finite_nat @ F2 )
     => ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2426_finite__imageI,axiom,
    ! [F2: set_int,H2: int > nat] :
      ( ( finite_finite_int @ F2 )
     => ( finite_finite_nat @ ( image_int_nat @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2427_finite__imageI,axiom,
    ! [F2: set_int,H2: int > int] :
      ( ( finite_finite_int @ F2 )
     => ( finite_finite_int @ ( image_int_int @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2428_finite__imageI,axiom,
    ! [F2: set_int,H2: int > complex] :
      ( ( finite_finite_int @ F2 )
     => ( finite3207457112153483333omplex @ ( image_int_complex @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2429_finite__imageI,axiom,
    ! [F2: set_int,H2: int > code_integer] :
      ( ( finite_finite_int @ F2 )
     => ( finite6017078050557962740nteger @ ( image_1587234942943678608nteger @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2430_finite__imageI,axiom,
    ! [F2: set_complex,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( finite_finite_nat @ ( image_complex_nat @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2431_finite__imageI,axiom,
    ! [F2: set_complex,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( finite_finite_int @ ( image_complex_int @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_2432_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_2433_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_2434_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_2435_insert__image,axiom,
    ! [X: nat,A4: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A4 ) )
        = ( image_nat_nat @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2436_insert__image,axiom,
    ! [X: nat,A4: set_nat,F: nat > vEBT_VEBT] :
      ( ( member_nat @ X @ A4 )
     => ( ( insert_VEBT_VEBT @ ( F @ X ) @ ( image_nat_VEBT_VEBT @ F @ A4 ) )
        = ( image_nat_VEBT_VEBT @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2437_insert__image,axiom,
    ! [X: nat,A4: set_nat,F: nat > int] :
      ( ( member_nat @ X @ A4 )
     => ( ( insert_int @ ( F @ X ) @ ( image_nat_int @ F @ A4 ) )
        = ( image_nat_int @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2438_insert__image,axiom,
    ! [X: nat,A4: set_nat,F: nat > $o] :
      ( ( member_nat @ X @ A4 )
     => ( ( insert_o @ ( F @ X ) @ ( image_nat_o @ F @ A4 ) )
        = ( image_nat_o @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2439_insert__image,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_VEBT_VEBT_nat @ F @ A4 ) )
        = ( image_VEBT_VEBT_nat @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2440_insert__image,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( insert_VEBT_VEBT @ ( F @ X ) @ ( image_3375948659692109573T_VEBT @ F @ A4 ) )
        = ( image_3375948659692109573T_VEBT @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2441_insert__image,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > int] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( insert_int @ ( F @ X ) @ ( image_VEBT_VEBT_int @ F @ A4 ) )
        = ( image_VEBT_VEBT_int @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2442_insert__image,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( insert_o @ ( F @ X ) @ ( image_VEBT_VEBT_o @ F @ A4 ) )
        = ( image_VEBT_VEBT_o @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2443_insert__image,axiom,
    ! [X: real,A4: set_real,F: real > nat] :
      ( ( member_real @ X @ A4 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_real_nat @ F @ A4 ) )
        = ( image_real_nat @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2444_insert__image,axiom,
    ! [X: real,A4: set_real,F: real > vEBT_VEBT] :
      ( ( member_real @ X @ A4 )
     => ( ( insert_VEBT_VEBT @ ( F @ X ) @ ( image_real_VEBT_VEBT @ F @ A4 ) )
        = ( image_real_VEBT_VEBT @ F @ A4 ) ) ) ).

% insert_image
thf(fact_2445_image__insert,axiom,
    ! [F: nat > nat,A: nat,B4: set_nat] :
      ( ( image_nat_nat @ F @ ( insert_nat @ A @ B4 ) )
      = ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2446_image__insert,axiom,
    ! [F: nat > vEBT_VEBT,A: nat,B4: set_nat] :
      ( ( image_nat_VEBT_VEBT @ F @ ( insert_nat @ A @ B4 ) )
      = ( insert_VEBT_VEBT @ ( F @ A ) @ ( image_nat_VEBT_VEBT @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2447_image__insert,axiom,
    ! [F: nat > int,A: nat,B4: set_nat] :
      ( ( image_nat_int @ F @ ( insert_nat @ A @ B4 ) )
      = ( insert_int @ ( F @ A ) @ ( image_nat_int @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2448_image__insert,axiom,
    ! [F: nat > $o,A: nat,B4: set_nat] :
      ( ( image_nat_o @ F @ ( insert_nat @ A @ B4 ) )
      = ( insert_o @ ( F @ A ) @ ( image_nat_o @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2449_image__insert,axiom,
    ! [F: vEBT_VEBT > nat,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( image_VEBT_VEBT_nat @ F @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( insert_nat @ ( F @ A ) @ ( image_VEBT_VEBT_nat @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2450_image__insert,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( image_3375948659692109573T_VEBT @ F @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( insert_VEBT_VEBT @ ( F @ A ) @ ( image_3375948659692109573T_VEBT @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2451_image__insert,axiom,
    ! [F: vEBT_VEBT > int,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( image_VEBT_VEBT_int @ F @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( insert_int @ ( F @ A ) @ ( image_VEBT_VEBT_int @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2452_image__insert,axiom,
    ! [F: vEBT_VEBT > $o,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( image_VEBT_VEBT_o @ F @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( insert_o @ ( F @ A ) @ ( image_VEBT_VEBT_o @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2453_image__insert,axiom,
    ! [F: int > nat,A: int,B4: set_int] :
      ( ( image_int_nat @ F @ ( insert_int @ A @ B4 ) )
      = ( insert_nat @ ( F @ A ) @ ( image_int_nat @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2454_image__insert,axiom,
    ! [F: int > vEBT_VEBT,A: int,B4: set_int] :
      ( ( image_int_VEBT_VEBT @ F @ ( insert_int @ A @ B4 ) )
      = ( insert_VEBT_VEBT @ ( F @ A ) @ ( image_int_VEBT_VEBT @ F @ B4 ) ) ) ).

% image_insert
thf(fact_2455_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_2456_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_2457_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_2458_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_2459_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_2460_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_2461_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_2462_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_2463_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_2464_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_2465_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_2466_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_2467_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_2468_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_2469_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_2470_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_2471_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_2472_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_2473_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_2474_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_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_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_2486_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_2487_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_2488_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_2489_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_2490_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_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_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_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_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_2505_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_2506_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_2507_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_2508_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_2509_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_2510_image__add__0,axiom,
    ! [S3: set_uint32] :
      ( ( image_uint32_uint32 @ ( plus_plus_uint32 @ zero_zero_uint32 ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_2511_image__add__0,axiom,
    ! [S3: set_real] :
      ( ( image_real_real @ ( plus_plus_real @ zero_zero_real ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_2512_image__add__0,axiom,
    ! [S3: set_rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_2513_image__add__0,axiom,
    ! [S3: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_2514_image__add__0,axiom,
    ! [S3: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_2515_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_2516_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_2517_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_2518_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_2519_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_2520_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_2521_enumerate__mono__iff,axiom,
    ! [S3: set_nat,M: nat,N: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M ) @ ( infini8530281810654367211te_nat @ S3 @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% enumerate_mono_iff
thf(fact_2522_add__right__imp__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_2523_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_2524_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_2525_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_2526_add__left__imp__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_2527_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_2528_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_2529_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_2530_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2531_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C2 ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2532_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2533_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2534_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A7: real,B7: real] : ( plus_plus_real @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2535_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A7: rat,B7: rat] : ( plus_plus_rat @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2536_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A7: nat,B7: nat] : ( plus_plus_nat @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2537_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A7: int,B7: int] : ( plus_plus_int @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2538_add_Oright__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_2539_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_2540_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_2541_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_2542_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_2543_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_2544_add_Oassoc,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_2545_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_2546_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_2547_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_2548_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B4 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2549_group__cancel_Oadd2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B4 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2550_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,B: nat,A: nat] :
      ( ( B4
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B4 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2551_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B4 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2552_group__cancel_Oadd1,axiom,
    ! [A4: real,K: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A4 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2553_group__cancel_Oadd1,axiom,
    ! [A4: rat,K: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A4 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2554_group__cancel_Oadd1,axiom,
    ! [A4: nat,K: nat,A: nat,B: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A4 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2555_group__cancel_Oadd1,axiom,
    ! [A4: int,K: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A4 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2556_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_2557_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_2558_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_2559_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_2560_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: nat,F: nat > nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2561_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: vEBT_VEBT,F: nat > vEBT_VEBT] :
      ( ( member_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_VEBT_VEBT @ B @ ( image_nat_VEBT_VEBT @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2562_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: real,F: nat > real] :
      ( ( member_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_real @ B @ ( image_nat_real @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2563_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: int,F: nat > int] :
      ( ( member_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_int @ B @ ( image_nat_int @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2564_rev__image__eqI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B: nat,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_VEBT_VEBT_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2565_rev__image__eqI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_VEBT_VEBT @ B @ ( image_3375948659692109573T_VEBT @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2566_rev__image__eqI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B: real,F: vEBT_VEBT > real] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_real @ B @ ( image_VEBT_VEBT_real @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2567_rev__image__eqI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,B: int,F: vEBT_VEBT > int] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_int @ B @ ( image_VEBT_VEBT_int @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2568_rev__image__eqI,axiom,
    ! [X: real,A4: set_real,B: nat,F: real > nat] :
      ( ( member_real @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_real_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2569_rev__image__eqI,axiom,
    ! [X: real,A4: set_real,B: vEBT_VEBT,F: real > vEBT_VEBT] :
      ( ( member_real @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_VEBT_VEBT @ B @ ( image_real_VEBT_VEBT @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_2570_ball__imageD,axiom,
    ! [F: nat > set_nat,A4: set_nat,P2: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_2571_ball__imageD,axiom,
    ! [F: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( image_nat_nat @ F @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_2572_ball__imageD,axiom,
    ! [F: nat > int,A4: set_nat,P2: int > $o] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( image_nat_int @ F @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_2573_ball__imageD,axiom,
    ! [F: int > nat,A4: set_int,P2: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( image_int_nat @ F @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_2574_ball__imageD,axiom,
    ! [F: int > int,A4: set_int,P2: int > $o] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( image_int_int @ F @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_2575_image__cong,axiom,
    ! [M7: set_nat,N8: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ( M7 = N8 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N8 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_set_nat @ F @ M7 )
          = ( image_nat_set_nat @ G @ N8 ) ) ) ) ).

% image_cong
thf(fact_2576_image__cong,axiom,
    ! [M7: set_nat,N8: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M7 = N8 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N8 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_nat @ F @ M7 )
          = ( image_nat_nat @ G @ N8 ) ) ) ) ).

% image_cong
thf(fact_2577_image__cong,axiom,
    ! [M7: set_nat,N8: set_nat,F: nat > int,G: nat > int] :
      ( ( M7 = N8 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N8 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_int @ F @ M7 )
          = ( image_nat_int @ G @ N8 ) ) ) ) ).

% image_cong
thf(fact_2578_image__cong,axiom,
    ! [M7: set_int,N8: set_int,F: int > nat,G: int > nat] :
      ( ( M7 = N8 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ N8 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_int_nat @ F @ M7 )
          = ( image_int_nat @ G @ N8 ) ) ) ) ).

% image_cong
thf(fact_2579_image__cong,axiom,
    ! [M7: set_int,N8: set_int,F: int > int,G: int > int] :
      ( ( M7 = N8 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ N8 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_int_int @ F @ M7 )
          = ( image_int_int @ G @ N8 ) ) ) ) ).

% image_cong
thf(fact_2580_bex__imageD,axiom,
    ! [F: nat > set_nat,A4: set_nat,P2: set_nat > $o] :
      ( ? [X5: set_nat] :
          ( ( member_set_nat @ X5 @ ( image_nat_set_nat @ F @ A4 ) )
          & ( P2 @ X5 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_2581_bex__imageD,axiom,
    ! [F: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( image_nat_nat @ F @ A4 ) )
          & ( P2 @ X5 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_2582_bex__imageD,axiom,
    ! [F: nat > int,A4: set_nat,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int @ X5 @ ( image_nat_int @ F @ A4 ) )
          & ( P2 @ X5 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_2583_bex__imageD,axiom,
    ! [F: int > nat,A4: set_int,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( image_int_nat @ F @ A4 ) )
          & ( P2 @ X5 ) )
     => ? [X3: int] :
          ( ( member_int @ X3 @ A4 )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_2584_bex__imageD,axiom,
    ! [F: int > int,A4: set_int,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int @ X5 @ ( image_int_int @ F @ A4 ) )
          & ( P2 @ X5 ) )
     => ? [X3: int] :
          ( ( member_int @ X3 @ A4 )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_2585_image__iff,axiom,
    ! [Z: nat,F: nat > nat,A4: set_nat] :
      ( ( member_nat @ Z @ ( image_nat_nat @ F @ A4 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_2586_image__iff,axiom,
    ! [Z: nat,F: int > nat,A4: set_int] :
      ( ( member_nat @ Z @ ( image_int_nat @ F @ A4 ) )
      = ( ? [X4: int] :
            ( ( member_int @ X4 @ A4 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_2587_image__iff,axiom,
    ! [Z: int,F: nat > int,A4: set_nat] :
      ( ( member_int @ Z @ ( image_nat_int @ F @ A4 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_2588_image__iff,axiom,
    ! [Z: int,F: int > int,A4: set_int] :
      ( ( member_int @ Z @ ( image_int_int @ F @ A4 ) )
      = ( ? [X4: int] :
            ( ( member_int @ X4 @ A4 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_2589_image__iff,axiom,
    ! [Z: set_nat,F: nat > set_nat,A4: set_nat] :
      ( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A4 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_2590_imageI,axiom,
    ! [X: nat,A4: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A4 )
     => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_2591_imageI,axiom,
    ! [X: nat,A4: set_nat,F: nat > vEBT_VEBT] :
      ( ( member_nat @ X @ A4 )
     => ( member_VEBT_VEBT @ ( F @ X ) @ ( image_nat_VEBT_VEBT @ F @ A4 ) ) ) ).

% imageI
thf(fact_2592_imageI,axiom,
    ! [X: nat,A4: set_nat,F: nat > real] :
      ( ( member_nat @ X @ A4 )
     => ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A4 ) ) ) ).

% imageI
thf(fact_2593_imageI,axiom,
    ! [X: nat,A4: set_nat,F: nat > int] :
      ( ( member_nat @ X @ A4 )
     => ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A4 ) ) ) ).

% imageI
thf(fact_2594_imageI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( member_nat @ ( F @ X ) @ ( image_VEBT_VEBT_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_2595_imageI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( member_VEBT_VEBT @ ( F @ X ) @ ( image_3375948659692109573T_VEBT @ F @ A4 ) ) ) ).

% imageI
thf(fact_2596_imageI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( member_real @ ( F @ X ) @ ( image_VEBT_VEBT_real @ F @ A4 ) ) ) ).

% imageI
thf(fact_2597_imageI,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > int] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( member_int @ ( F @ X ) @ ( image_VEBT_VEBT_int @ F @ A4 ) ) ) ).

% imageI
thf(fact_2598_imageI,axiom,
    ! [X: real,A4: set_real,F: real > nat] :
      ( ( member_real @ X @ A4 )
     => ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_2599_imageI,axiom,
    ! [X: real,A4: set_real,F: real > vEBT_VEBT] :
      ( ( member_real @ X @ A4 )
     => ( member_VEBT_VEBT @ ( F @ X ) @ ( image_real_VEBT_VEBT @ F @ A4 ) ) ) ).

% imageI
thf(fact_2600_image__mono,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A4 ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_2601_image__mono,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_2602_image__mono,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > int] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B4 ) ) ) ).

% image_mono
thf(fact_2603_image__mono,axiom,
    ! [A4: set_int,B4: set_int,F: int > nat] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_2604_image__mono,axiom,
    ! [A4: set_int,B4: set_int,F: int > int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B4 ) ) ) ).

% image_mono
thf(fact_2605_image__subsetI,axiom,
    ! [A4: set_nat,F: nat > nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2606_image__subsetI,axiom,
    ! [A4: set_nat,F: nat > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_nat_VEBT_VEBT @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2607_image__subsetI,axiom,
    ! [A4: set_nat,F: nat > real,B4: set_real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_nat_real @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2608_image__subsetI,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > nat,B4: set_nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_VEBT_VEBT_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2609_image__subsetI,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_3375948659692109573T_VEBT @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2610_image__subsetI,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real,B4: set_real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_VEBT_VEBT_real @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2611_image__subsetI,axiom,
    ! [A4: set_real,F: real > nat,B4: set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2612_image__subsetI,axiom,
    ! [A4: set_real,F: real > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_real_VEBT_VEBT @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2613_image__subsetI,axiom,
    ! [A4: set_real,F: real > real,B4: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_real_real @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2614_image__subsetI,axiom,
    ! [A4: set_int,F: int > nat,B4: set_nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_2615_subset__imageE,axiom,
    ! [B4: set_set_nat,F: nat > set_nat,A4: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A4 ) )
     => ~ ! [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
           => ( B4
             != ( image_nat_set_nat @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_2616_subset__imageE,axiom,
    ! [B4: set_nat,F: nat > nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
     => ~ ! [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
           => ( B4
             != ( image_nat_nat @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_2617_subset__imageE,axiom,
    ! [B4: set_nat,F: int > nat,A4: set_int] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A4 ) )
     => ~ ! [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A4 )
           => ( B4
             != ( image_int_nat @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_2618_subset__imageE,axiom,
    ! [B4: set_int,F: nat > int,A4: set_nat] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A4 ) )
     => ~ ! [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
           => ( B4
             != ( image_nat_int @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_2619_subset__imageE,axiom,
    ! [B4: set_int,F: int > int,A4: set_int] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A4 ) )
     => ~ ! [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A4 )
           => ( B4
             != ( image_int_int @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_2620_image__subset__iff,axiom,
    ! [F: nat > nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( member_nat @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_2621_image__subset__iff,axiom,
    ! [F: int > nat,A4: set_int,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B4 )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A4 )
           => ( member_nat @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_2622_image__subset__iff,axiom,
    ! [F: nat > set_nat,A4: set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A4 ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( member_set_nat @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_2623_image__subset__iff,axiom,
    ! [F: nat > int,A4: set_nat,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( member_int @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_2624_image__subset__iff,axiom,
    ! [F: int > int,A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B4 )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A4 )
           => ( member_int @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_2625_subset__image__iff,axiom,
    ! [B4: set_set_nat,F: nat > set_nat,A4: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A4 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A4 )
            & ( B4
              = ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_2626_subset__image__iff,axiom,
    ! [B4: set_nat,F: nat > nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A4 )
            & ( B4
              = ( image_nat_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_2627_subset__image__iff,axiom,
    ! [B4: set_nat,F: int > nat,A4: set_int] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A4 ) )
      = ( ? [AA: set_int] :
            ( ( ord_less_eq_set_int @ AA @ A4 )
            & ( B4
              = ( image_int_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_2628_subset__image__iff,axiom,
    ! [B4: set_int,F: nat > int,A4: set_nat] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A4 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A4 )
            & ( B4
              = ( image_nat_int @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_2629_subset__image__iff,axiom,
    ! [B4: set_int,F: int > int,A4: set_int] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A4 ) )
      = ( ? [AA: set_int] :
            ( ( ord_less_eq_set_int @ AA @ A4 )
            & ( B4
              = ( image_int_int @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_2630_all__subset__image,axiom,
    ! [F: nat > set_nat,A4: set_nat,P2: set_set_nat > $o] :
      ( ( ! [B6: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F @ A4 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ A4 )
           => ( P2 @ ( image_nat_set_nat @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_2631_all__subset__image,axiom,
    ! [F: nat > nat,A4: set_nat,P2: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ A4 )
           => ( P2 @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_2632_all__subset__image,axiom,
    ! [F: int > nat,A4: set_int,P2: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A4 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ A4 )
           => ( P2 @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_2633_all__subset__image,axiom,
    ! [F: nat > int,A4: set_nat,P2: set_int > $o] :
      ( ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A4 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ A4 )
           => ( P2 @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_2634_all__subset__image,axiom,
    ! [F: int > int,A4: set_int,P2: set_int > $o] :
      ( ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A4 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ A4 )
           => ( P2 @ ( image_int_int @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_2635_verit__sum__simplify,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% verit_sum_simplify
thf(fact_2636_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_2637_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_2638_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2639_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_2640_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_2641_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_2642_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_2643_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_2644_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_2645_add_Ocomm__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% add.comm_neutral
thf(fact_2646_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2647_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2648_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2649_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2650_add_Ogroup__left__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2651_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2652_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2653_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2654_add__le__imp__le__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2655_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2656_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2657_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2658_add__le__imp__le__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2659_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2660_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2661_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2662_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] :
        ? [C6: nat] :
          ( B7
          = ( plus_plus_nat @ A7 @ C6 ) ) ) ) ).

% le_iff_add
thf(fact_2663_add__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_2664_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_2665_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_2666_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_2667_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C: nat] :
            ( B
           != ( plus_plus_nat @ A @ C ) ) ) ).

% less_eqE
thf(fact_2668_add__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_2669_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_2670_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_2671_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_2672_add__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2673_add__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2674_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2675_add__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2676_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_2677_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_2678_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_2679_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_2680_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_2681_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_2682_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_2683_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_2684_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_2685_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_2686_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_2687_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_2688_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_2689_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_2690_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_2691_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_2692_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_2693_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_2694_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_2695_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_2696_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_2697_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_2698_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_2699_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_2700_add__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2701_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2702_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2703_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2704_add__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2705_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2706_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2707_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2708_add__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2709_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2710_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2711_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2712_add__less__imp__less__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2713_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2714_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2715_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2716_add__less__imp__less__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2717_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2718_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2719_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2720_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2721_diff__diff__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( minus_minus_uint32 @ ( minus_minus_uint32 @ A @ B ) @ C2 )
      = ( minus_minus_uint32 @ A @ ( plus_plus_uint32 @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2722_diff__diff__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2723_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2724_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2725_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2726_add__implies__diff,axiom,
    ! [C2: uint32,B: uint32,A: uint32] :
      ( ( ( plus_plus_uint32 @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_uint32 @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2727_add__implies__diff,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2728_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2729_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2730_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2731_diff__add__eq__diff__diff__swap,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( minus_minus_uint32 @ A @ ( plus_plus_uint32 @ B @ C2 ) )
      = ( minus_minus_uint32 @ ( minus_minus_uint32 @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2732_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2733_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2734_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_2735_diff__add__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( plus_plus_uint32 @ ( minus_minus_uint32 @ A @ B ) @ C2 )
      = ( minus_minus_uint32 @ ( plus_plus_uint32 @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_2736_diff__add__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_2737_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_2738_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2739_diff__diff__eq2,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( minus_minus_uint32 @ A @ ( minus_minus_uint32 @ B @ C2 ) )
      = ( minus_minus_uint32 @ ( plus_plus_uint32 @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2740_diff__diff__eq2,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2741_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2742_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_2743_add__diff__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( plus_plus_uint32 @ A @ ( minus_minus_uint32 @ B @ C2 ) )
      = ( minus_minus_uint32 @ ( plus_plus_uint32 @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_2744_add__diff__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_2745_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_2746_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C2 @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_2747_eq__diff__eq,axiom,
    ! [A: uint32,C2: uint32,B: uint32] :
      ( ( A
        = ( minus_minus_uint32 @ C2 @ B ) )
      = ( ( plus_plus_uint32 @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_2748_eq__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C2 @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_2749_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C2 @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_2750_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_2751_diff__eq__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( ( minus_minus_uint32 @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_uint32 @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_2752_diff__eq__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_real @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_2753_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_2754_group__cancel_Osub1,axiom,
    ! [A4: rat,K: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A4 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2755_group__cancel_Osub1,axiom,
    ! [A4: uint32,K: uint32,A: uint32,B: uint32] :
      ( ( A4
        = ( plus_plus_uint32 @ K @ A ) )
     => ( ( minus_minus_uint32 @ A4 @ B )
        = ( plus_plus_uint32 @ K @ ( minus_minus_uint32 @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2756_group__cancel_Osub1,axiom,
    ! [A4: real,K: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A4 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2757_group__cancel_Osub1,axiom,
    ! [A4: int,K: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A4 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2758_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_2759_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_2760_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_2761_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_2762_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_2763_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_2764_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_2765_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_2766_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_2767_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_2768_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_2769_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_2770_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_2771_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N6: nat] :
        ? [K3: nat] :
          ( N6
          = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_2772_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_2773_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_2774_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_2775_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_2776_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_2777_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_2778_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_2779_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_2780_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_2781_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_2782_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_2783_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_2784_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_2785_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_2786_enumerate__in__set,axiom,
    ! [S3: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( member_nat @ ( infini8530281810654367211te_nat @ S3 @ N ) @ S3 ) ) ).

% enumerate_in_set
thf(fact_2787_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ( plus_plus_real @ A @ A )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2788_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ( plus_plus_rat @ A @ A )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2789_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ( plus_plus_int @ A @ A )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2790_enumerate__Ex,axiom,
    ! [S3: set_nat,S: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ( member_nat @ S @ S3 )
       => ? [N2: nat] :
            ( ( infini8530281810654367211te_nat @ S3 @ N2 )
            = S ) ) ) ).

% enumerate_Ex
thf(fact_2791_all__finite__subset__image,axiom,
    ! [F: nat > nat,A4: set_nat,P2: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A4 ) )
           => ( P2 @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2792_all__finite__subset__image,axiom,
    ! [F: complex > nat,A4: set_complex,P2: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_complex_nat @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B6 )
              & ( ord_le211207098394363844omplex @ B6 @ A4 ) )
           => ( P2 @ ( image_complex_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2793_all__finite__subset__image,axiom,
    ! [F: code_integer > nat,A4: set_Code_integer,P2: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_951025933927791156er_nat @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ A4 ) )
           => ( P2 @ ( image_951025933927791156er_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2794_all__finite__subset__image,axiom,
    ! [F: nat > complex,A4: set_nat,P2: set_complex > $o] :
      ( ( ! [B6: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B6 )
              & ( ord_le211207098394363844omplex @ B6 @ ( image_nat_complex @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A4 ) )
           => ( P2 @ ( image_nat_complex @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2795_all__finite__subset__image,axiom,
    ! [F: complex > complex,A4: set_complex,P2: set_complex > $o] :
      ( ( ! [B6: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B6 )
              & ( ord_le211207098394363844omplex @ B6 @ ( image_1468599708987790691omplex @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B6 )
              & ( ord_le211207098394363844omplex @ B6 @ A4 ) )
           => ( P2 @ ( image_1468599708987790691omplex @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2796_all__finite__subset__image,axiom,
    ! [F: code_integer > complex,A4: set_Code_integer,P2: set_complex > $o] :
      ( ( ! [B6: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B6 )
              & ( ord_le211207098394363844omplex @ B6 @ ( image_3397630267976458002omplex @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ A4 ) )
           => ( P2 @ ( image_3397630267976458002omplex @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2797_all__finite__subset__image,axiom,
    ! [F: nat > code_integer,A4: set_nat,P2: set_Code_integer > $o] :
      ( ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ ( image_1215581382706833972nteger @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A4 ) )
           => ( P2 @ ( image_1215581382706833972nteger @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2798_all__finite__subset__image,axiom,
    ! [F: complex > code_integer,A4: set_complex,P2: set_Code_integer > $o] :
      ( ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ ( image_1994230757181692690nteger @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B6 )
              & ( ord_le211207098394363844omplex @ B6 @ A4 ) )
           => ( P2 @ ( image_1994230757181692690nteger @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2799_all__finite__subset__image,axiom,
    ! [F: code_integer > code_integer,A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ ( image_4470545334726330049nteger @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ A4 ) )
           => ( P2 @ ( image_4470545334726330049nteger @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2800_all__finite__subset__image,axiom,
    ! [F: int > nat,A4: set_int,P2: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A4 ) ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ( finite_finite_int @ B6 )
              & ( ord_less_eq_set_int @ B6 @ A4 ) )
           => ( P2 @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_2801_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A4: set_nat,P2: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A4 )
            & ( P2 @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2802_ex__finite__subset__image,axiom,
    ! [F: complex > nat,A4: set_complex,P2: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_complex_nat @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_complex] :
            ( ( finite3207457112153483333omplex @ B6 )
            & ( ord_le211207098394363844omplex @ B6 @ A4 )
            & ( P2 @ ( image_complex_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2803_ex__finite__subset__image,axiom,
    ! [F: code_integer > nat,A4: set_Code_integer,P2: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_951025933927791156er_nat @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ A4 )
            & ( P2 @ ( image_951025933927791156er_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2804_ex__finite__subset__image,axiom,
    ! [F: nat > complex,A4: set_nat,P2: set_complex > $o] :
      ( ( ? [B6: set_complex] :
            ( ( finite3207457112153483333omplex @ B6 )
            & ( ord_le211207098394363844omplex @ B6 @ ( image_nat_complex @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A4 )
            & ( P2 @ ( image_nat_complex @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2805_ex__finite__subset__image,axiom,
    ! [F: complex > complex,A4: set_complex,P2: set_complex > $o] :
      ( ( ? [B6: set_complex] :
            ( ( finite3207457112153483333omplex @ B6 )
            & ( ord_le211207098394363844omplex @ B6 @ ( image_1468599708987790691omplex @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_complex] :
            ( ( finite3207457112153483333omplex @ B6 )
            & ( ord_le211207098394363844omplex @ B6 @ A4 )
            & ( P2 @ ( image_1468599708987790691omplex @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2806_ex__finite__subset__image,axiom,
    ! [F: code_integer > complex,A4: set_Code_integer,P2: set_complex > $o] :
      ( ( ? [B6: set_complex] :
            ( ( finite3207457112153483333omplex @ B6 )
            & ( ord_le211207098394363844omplex @ B6 @ ( image_3397630267976458002omplex @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ A4 )
            & ( P2 @ ( image_3397630267976458002omplex @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2807_ex__finite__subset__image,axiom,
    ! [F: nat > code_integer,A4: set_nat,P2: set_Code_integer > $o] :
      ( ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ ( image_1215581382706833972nteger @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A4 )
            & ( P2 @ ( image_1215581382706833972nteger @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2808_ex__finite__subset__image,axiom,
    ! [F: complex > code_integer,A4: set_complex,P2: set_Code_integer > $o] :
      ( ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ ( image_1994230757181692690nteger @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_complex] :
            ( ( finite3207457112153483333omplex @ B6 )
            & ( ord_le211207098394363844omplex @ B6 @ A4 )
            & ( P2 @ ( image_1994230757181692690nteger @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2809_ex__finite__subset__image,axiom,
    ! [F: code_integer > code_integer,A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ ( image_4470545334726330049nteger @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ A4 )
            & ( P2 @ ( image_4470545334726330049nteger @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2810_ex__finite__subset__image,axiom,
    ! [F: int > nat,A4: set_int,P2: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A4 ) )
            & ( P2 @ B6 ) ) )
      = ( ? [B6: set_int] :
            ( ( finite_finite_int @ B6 )
            & ( ord_less_eq_set_int @ B6 @ A4 )
            & ( P2 @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_2811_finite__subset__image,axiom,
    ! [B4: set_nat,F: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
            & ( finite_finite_nat @ C5 )
            & ( B4
              = ( image_nat_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2812_finite__subset__image,axiom,
    ! [B4: set_nat,F: complex > nat,A4: set_complex] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_complex_nat @ F @ A4 ) )
       => ? [C5: set_complex] :
            ( ( ord_le211207098394363844omplex @ C5 @ A4 )
            & ( finite3207457112153483333omplex @ C5 )
            & ( B4
              = ( image_complex_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2813_finite__subset__image,axiom,
    ! [B4: set_nat,F: code_integer > nat,A4: set_Code_integer] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_951025933927791156er_nat @ F @ A4 ) )
       => ? [C5: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C5 @ A4 )
            & ( finite6017078050557962740nteger @ C5 )
            & ( B4
              = ( image_951025933927791156er_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2814_finite__subset__image,axiom,
    ! [B4: set_complex,F: nat > complex,A4: set_nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_nat_complex @ F @ A4 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
            & ( finite_finite_nat @ C5 )
            & ( B4
              = ( image_nat_complex @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2815_finite__subset__image,axiom,
    ! [B4: set_complex,F: complex > complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_1468599708987790691omplex @ F @ A4 ) )
       => ? [C5: set_complex] :
            ( ( ord_le211207098394363844omplex @ C5 @ A4 )
            & ( finite3207457112153483333omplex @ C5 )
            & ( B4
              = ( image_1468599708987790691omplex @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2816_finite__subset__image,axiom,
    ! [B4: set_complex,F: code_integer > complex,A4: set_Code_integer] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_3397630267976458002omplex @ F @ A4 ) )
       => ? [C5: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C5 @ A4 )
            & ( finite6017078050557962740nteger @ C5 )
            & ( B4
              = ( image_3397630267976458002omplex @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2817_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: nat > code_integer,A4: set_nat] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A4 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
            & ( finite_finite_nat @ C5 )
            & ( B4
              = ( image_1215581382706833972nteger @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2818_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: complex > code_integer,A4: set_complex] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1994230757181692690nteger @ F @ A4 ) )
       => ? [C5: set_complex] :
            ( ( ord_le211207098394363844omplex @ C5 @ A4 )
            & ( finite3207457112153483333omplex @ C5 )
            & ( B4
              = ( image_1994230757181692690nteger @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2819_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: code_integer > code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A4 ) )
       => ? [C5: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C5 @ A4 )
            & ( finite6017078050557962740nteger @ C5 )
            & ( B4
              = ( image_4470545334726330049nteger @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2820_finite__subset__image,axiom,
    ! [B4: set_nat,F: int > nat,A4: set_int] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A4 ) )
       => ? [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A4 )
            & ( finite_finite_int @ C5 )
            & ( B4
              = ( image_int_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_2821_finite__surj,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_2822_finite__surj,axiom,
    ! [A4: set_nat,B4: set_complex,F: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_nat_complex @ F @ A4 ) )
       => ( finite3207457112153483333omplex @ B4 ) ) ) ).

% finite_surj
thf(fact_2823_finite__surj,axiom,
    ! [A4: set_nat,B4: set_Code_integer,F: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A4 ) )
       => ( finite6017078050557962740nteger @ B4 ) ) ) ).

% finite_surj
thf(fact_2824_finite__surj,axiom,
    ! [A4: set_int,B4: set_nat,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A4 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_2825_finite__surj,axiom,
    ! [A4: set_int,B4: set_complex,F: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_int_complex @ F @ A4 ) )
       => ( finite3207457112153483333omplex @ B4 ) ) ) ).

% finite_surj
thf(fact_2826_finite__surj,axiom,
    ! [A4: set_int,B4: set_Code_integer,F: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1587234942943678608nteger @ F @ A4 ) )
       => ( finite6017078050557962740nteger @ B4 ) ) ) ).

% finite_surj
thf(fact_2827_finite__surj,axiom,
    ! [A4: set_complex,B4: set_nat,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_complex_nat @ F @ A4 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_2828_finite__surj,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_1468599708987790691omplex @ F @ A4 ) )
       => ( finite3207457112153483333omplex @ B4 ) ) ) ).

% finite_surj
thf(fact_2829_finite__surj,axiom,
    ! [A4: set_complex,B4: set_Code_integer,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1994230757181692690nteger @ F @ A4 ) )
       => ( finite6017078050557962740nteger @ B4 ) ) ) ).

% finite_surj
thf(fact_2830_finite__surj,axiom,
    ! [A4: set_Code_integer,B4: set_nat,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_951025933927791156er_nat @ F @ A4 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_2831_image__diff__subset,axiom,
    ! [F: nat > set_nat,A4: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A4 ) @ ( image_nat_set_nat @ F @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_2832_image__diff__subset,axiom,
    ! [F: int > nat,A4: set_int,B4: set_int] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B4 ) ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_2833_image__diff__subset,axiom,
    ! [F: nat > nat,A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B4 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_2834_image__diff__subset,axiom,
    ! [F: int > int,A4: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B4 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_2835_image__diff__subset,axiom,
    ! [F: nat > int,A4: set_nat,B4: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B4 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_2836_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2837_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2838_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2839_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2840_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2841_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2842_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2843_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2844_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_2845_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_2846_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_2847_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_2848_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_2849_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_2850_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_2851_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_2852_add__increasing2,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2853_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2854_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2855_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2856_add__decreasing2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2857_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2858_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2859_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2860_add__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C2 )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2861_add__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2862_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2863_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2864_add__decreasing,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C2 @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2865_add__decreasing,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2866_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2867_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2868_pos__add__strict,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2869_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2870_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2871_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2872_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2873_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_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_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_2881_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2882_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2883_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_2884_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_2885_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_2886_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_2887_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_2888_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_2889_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_2890_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_2891_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_2892_add__le__less__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2893_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2894_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2895_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2896_add__less__le__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2897_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2898_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2899_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2900_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_2901_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_2902_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_2903_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_2904_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2905_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2906_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2907_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2908_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C2 )
          = ( B
            = ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_2909_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_2910_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_2911_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_2912_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_2913_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
        = ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_2914_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_2915_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_2916_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% le_add_diff
thf(fact_2917_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_2918_le__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C2 @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2919_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2920_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2921_diff__le__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_2922_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_2923_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_2924_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2925_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2926_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2927_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2928_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2929_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2930_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2931_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2932_diff__less__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( ord_less_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_2933_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_2934_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_2935_less__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C2 @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2936_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2937_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2938_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_2939_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_2940_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_2941_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_2942_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
       != zero_zero_real ) ) ).

% Ints_odd_nonzero
thf(fact_2943_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_2944_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_2945_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_2946_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_2947_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q5: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q5 ) ) ) ) ).

% less_natE
thf(fact_2948_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_2949_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_2950_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N6: nat] :
        ? [K3: nat] :
          ( N6
          = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_2951_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_2952_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2953_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_2954_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_2955_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N6: nat] : ( plus_plus_nat @ N6 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_2956_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_2957_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_2958_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_2959_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_2960_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_2961_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_2962_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_2963_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_2964_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_2965_le__enumerate,axiom,
    ! [S3: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ).

% le_enumerate
thf(fact_2966_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: nat > set_nat,X: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A4 )
           => ( ( F @ Y3 )
              = ( F @ X ) ) )
       => ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A4 ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_2967_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: nat > nat,X: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A4 )
           => ( ( F @ Y3 )
              = ( F @ X ) ) )
       => ( ( the_elem_nat @ ( image_nat_nat @ F @ A4 ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_2968_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: nat > int,X: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A4 )
           => ( ( F @ Y3 )
              = ( F @ X ) ) )
       => ( ( the_elem_int @ ( image_nat_int @ F @ A4 ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_2969_the__elem__image__unique,axiom,
    ! [A4: set_int,F: int > nat,X: int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ A4 )
           => ( ( F @ Y3 )
              = ( F @ X ) ) )
       => ( ( the_elem_nat @ ( image_int_nat @ F @ A4 ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_2970_the__elem__image__unique,axiom,
    ! [A4: set_int,F: int > int,X: int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ A4 )
           => ( ( F @ Y3 )
              = ( F @ X ) ) )
       => ( ( the_elem_int @ ( image_int_int @ F @ A4 ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_2971_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
        = ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% Ints_odd_less_0
thf(fact_2972_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
        = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_2973_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_2974_add__strict__increasing2,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2975_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2976_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2977_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2978_add__strict__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C2 )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2979_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2980_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2981_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2982_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_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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_2990_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_2991_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_2992_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_2993_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_2994_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_2995_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_2996_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_2997_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_2998_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2999_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3000_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3001_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3002_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A7 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3003_discrete,axiom,
    ( ord_less_int
    = ( ^ [A7: int] : ( ord_less_eq_int @ ( plus_plus_int @ A7 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3004_nat__diff__split__asm,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P2 @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D5 ) )
                & ~ ( P2 @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3005_nat__diff__split,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P2 @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D5 ) )
           => ( P2 @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_3006_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_3007_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu2 @ 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_3008_enumerate__step,axiom,
    ! [S3: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ N ) @ ( infini8530281810654367211te_nat @ S3 @ ( suc @ N ) ) ) ) ).

% enumerate_step
thf(fact_3009_enumerate__mono,axiom,
    ! [M: nat,N: nat,S3: set_nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ~ ( finite_finite_nat @ S3 )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M ) @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ) ).

% enumerate_mono
thf(fact_3010_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N6: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N6 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N6 ) ) ) ) ) ).

% add_eq_if
thf(fact_3011_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_3012_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_3013_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( 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_3014_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,Va2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
      = ( 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_3015_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu2 @ 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_3016_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ Uw2 ) @ ( 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_3017_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% pred_bound_height'
thf(fact_3018_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% succ'_bound_height
thf(fact_3019_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% insert'_bound_height
thf(fact_3020_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% member_bound_height'
thf(fact_3021_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( 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 @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_3022_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).

% max_ins_scaled
thf(fact_3023_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3024_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3025_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ! [A3: $o] :
            ( ? [B3: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_3026_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_3027_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( 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 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_3028_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ 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 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_3029_Euclid__induct,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 @ A3 @ B3 )
          = ( P2 @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P2 @ A3 @ B3 )
             => ( P2 @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P2 @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_3030_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_3031_mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C2 )
        = ( times_times_rat @ B @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_3032_mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ( times_times_real @ A @ C2 )
        = ( times_times_real @ B @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_3033_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C2 )
        = ( times_times_nat @ B @ C2 ) )
      = ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_3034_mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ( times_times_int @ A @ C2 )
        = ( times_times_int @ B @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_3035_mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C2 @ A )
        = ( times_times_rat @ C2 @ B ) )
      = ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_3036_mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ( times_times_real @ C2 @ A )
        = ( times_times_real @ C2 @ B ) )
      = ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_3037_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C2 @ A )
        = ( times_times_nat @ C2 @ B ) )
      = ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_3038_mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ( times_times_int @ C2 @ A )
        = ( times_times_int @ C2 @ B ) )
      = ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_3039_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_3040_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_3041_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_3042_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_3043_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_3044_mult__zero__right,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ zero_zero_uint32 )
      = zero_zero_uint32 ) ).

% mult_zero_right
thf(fact_3045_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_3046_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_3047_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_3048_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_3049_mult__zero__left,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ zero_zero_uint32 @ A )
      = zero_zero_uint32 ) ).

% mult_zero_left
thf(fact_3050_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_3051_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_3052_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_3053_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_3054_mult_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.right_neutral
thf(fact_3055_mult_Oright__neutral,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ one_one_uint32 )
      = A ) ).

% mult.right_neutral
thf(fact_3056_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_3057_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_3058_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_3059_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_3060_mult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% mult_1
thf(fact_3061_mult__1,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ one_one_uint32 @ A )
      = A ) ).

% mult_1
thf(fact_3062_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_3063_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_3064_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_3065_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_3066_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_3067_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_3068_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_3069_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_3070_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_3071_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_3072_mult__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ( times_times_rat @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_3073_mult__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( ( times_times_real @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_3074_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ( times_times_int @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_3075_mult__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2
        = ( times_times_rat @ B @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_3076_mult__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( C2
        = ( times_times_real @ B @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_3077_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2
        = ( times_times_int @ B @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_3078_mult__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ( times_times_rat @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_3079_mult__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( ( times_times_real @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_3080_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ( times_times_int @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_3081_mult__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2
        = ( times_times_rat @ C2 @ B ) )
      = ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_3082_mult__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( C2
        = ( times_times_real @ C2 @ B ) )
      = ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_3083_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2
        = ( times_times_int @ C2 @ B ) )
      = ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_3084_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_3085_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_3086_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_3087_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_3088_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_3089_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_3090_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_3091_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C2 )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_3092_mult_Oassoc,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( times_times_uint32 @ ( times_times_uint32 @ A @ B ) @ C2 )
      = ( times_times_uint32 @ A @ ( times_times_uint32 @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_3093_mult_Oassoc,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
      = ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_3094_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_3095_mult_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
      = ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_3096_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A7: assn,B7: assn] : ( times_times_assn @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3097_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_uint32
    = ( ^ [A7: uint32,B7: uint32] : ( times_times_uint32 @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3098_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A7: real,B7: real] : ( times_times_real @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3099_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A7: nat,B7: nat] : ( times_times_nat @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3100_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A7: int,B7: int] : ( times_times_int @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_3101_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( times_times_assn @ B @ ( times_times_assn @ A @ C2 ) )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3102_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: uint32,A: uint32,C2: uint32] :
      ( ( times_times_uint32 @ B @ ( times_times_uint32 @ A @ C2 ) )
      = ( times_times_uint32 @ A @ ( times_times_uint32 @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3103_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3104_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3105_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_3106_mult__right__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C2 )
          = ( times_times_rat @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3107_mult__right__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ A @ C2 )
          = ( times_times_real @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3108_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C2 )
          = ( times_times_nat @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3109_mult__right__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ A @ C2 )
          = ( times_times_int @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3110_mult__left__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ C2 @ A )
          = ( times_times_rat @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3111_mult__left__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ C2 @ A )
          = ( times_times_real @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3112_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ C2 @ A )
          = ( times_times_nat @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3113_mult__left__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ C2 @ A )
          = ( times_times_int @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3114_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_3115_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_3116_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_3117_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_3118_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_3119_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_3120_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_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3129_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ one_one_uint32 @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3130_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_3131_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_3132_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_3133_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_3134_mult_Ocomm__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.comm_neutral
thf(fact_3135_mult_Ocomm__neutral,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ one_one_uint32 )
      = A ) ).

% mult.comm_neutral
thf(fact_3136_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_3137_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_3138_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_3139_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_3140_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_3141_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_3142_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_3143_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_3144_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_3145_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_3146_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_3147_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_3148_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_3149_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_3150_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_3151_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_3152_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_3153_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3154_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3155_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3156_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3157_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_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_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_3165_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_3166_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_3167_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_3168_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_3169_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_3170_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_3171_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_3172_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_3173_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_3174_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_3175_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_3176_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_3177_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_3178_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_3179_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_3180_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_3181_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_3182_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_3183_mult__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_3184_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_3185_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_3186_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_3187_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3188_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3189_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3190_mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3191_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3192_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3193_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3194_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_3195_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_3196_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_3197_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3198_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3199_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3200_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_3201_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_3202_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_3203_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_3204_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_3205_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_3206_mult__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3207_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3208_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3209_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3210_mult__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3211_mult__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3212_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3213_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3214_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_3215_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_3216_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_3217_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_3218_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_3219_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_3220_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_3221_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_3222_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_3223_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_3224_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_3225_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_3226_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_3227_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_3228_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_3229_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_3230_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_3231_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_3232_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_3233_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_3234_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_3235_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_3236_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_3237_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_3238_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_3239_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_3240_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_3241_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_3242_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_3243_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_3244_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_3245_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_3246_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_3247_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_3248_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_3249_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_3250_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3251_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3252_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3253_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3254_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3255_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3256_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3257_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3258_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3259_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3260_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3261_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3262_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3263_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C2 @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3264_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3265_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3266_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3267_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3268_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3269_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3270_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3271_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3272_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3273_mult__less__cancel__right__disj,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C2 @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3274_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3275_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3276_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3277_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3278_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3279_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3280_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_3281_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_3282_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_3283_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_3284_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_3285_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_3286_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_3287_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_3288_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_3289_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_3290_mlex__snd__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N8: nat] :
      ( ( A = A2 )
     => ( ( ord_less_nat @ B @ B2 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N8 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N8 ) @ B2 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_3291_mlex__fst__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,N8: nat,B2: nat] :
      ( ( ord_less_nat @ A @ A2 )
     => ( ( ord_less_nat @ B @ N8 )
       => ( ( ord_less_nat @ B2 @ N8 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N8 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N8 ) @ B2 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_3292_mlex__bound,axiom,
    ! [A: nat,A4: nat,B: nat,N8: nat] :
      ( ( ord_less_nat @ A @ A4 )
     => ( ( ord_less_nat @ B @ N8 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N8 ) @ B ) @ ( times_times_nat @ A4 @ N8 ) ) ) ) ).

% mlex_bound
thf(fact_3293_mlex__leI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N8: nat] :
      ( ( ord_less_eq_nat @ A @ A2 )
     => ( ( ord_less_eq_nat @ B @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N8 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N8 ) @ B2 ) ) ) ) ).

% mlex_leI
thf(fact_3294_field__le__mult__one__interval,axiom,
    ! [X: real,Y: 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 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3295_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: 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 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3296_in__image__insert__iff,axiom,
    ! [B4: set_set_VEBT_VEBT,X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ! [C5: set_VEBT_VEBT] :
          ( ( member_set_VEBT_VEBT @ C5 @ B4 )
         => ~ ( member_VEBT_VEBT @ X @ C5 ) )
     => ( ( member_set_VEBT_VEBT @ A4 @ ( image_1661326939266726661T_VEBT @ ( insert_VEBT_VEBT @ X ) @ B4 ) )
        = ( ( member_VEBT_VEBT @ X @ A4 )
          & ( member_set_VEBT_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_3297_in__image__insert__iff,axiom,
    ! [B4: set_set_real,X: real,A4: set_real] :
      ( ! [C5: set_real] :
          ( ( member_set_real @ C5 @ B4 )
         => ~ ( member_real @ X @ C5 ) )
     => ( ( member_set_real @ A4 @ ( image_2436557299294012491t_real @ ( insert_real @ X ) @ B4 ) )
        = ( ( member_real @ X @ A4 )
          & ( member_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_3298_in__image__insert__iff,axiom,
    ! [B4: set_set_set_nat,X: set_nat,A4: set_set_nat] :
      ( ! [C5: set_set_nat] :
          ( ( member_set_set_nat @ C5 @ B4 )
         => ~ ( member_set_nat @ X @ C5 ) )
     => ( ( member_set_set_nat @ A4 @ ( image_7884819252390400639et_nat @ ( insert_set_nat @ X ) @ B4 ) )
        = ( ( member_set_nat @ X @ A4 )
          & ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_3299_in__image__insert__iff,axiom,
    ! [B4: set_set_int,X: int,A4: set_int] :
      ( ! [C5: set_int] :
          ( ( member_set_int @ C5 @ B4 )
         => ~ ( member_int @ X @ C5 ) )
     => ( ( member_set_int @ A4 @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B4 ) )
        = ( ( member_int @ X @ A4 )
          & ( member_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_3300_in__image__insert__iff,axiom,
    ! [B4: set_set_o,X: $o,A4: set_o] :
      ( ! [C5: set_o] :
          ( ( member_set_o @ C5 @ B4 )
         => ~ ( member_o @ X @ C5 ) )
     => ( ( member_set_o @ A4 @ ( image_set_o_set_o @ ( insert_o @ X ) @ B4 ) )
        = ( ( member_o @ X @ A4 )
          & ( member_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_3301_in__image__insert__iff,axiom,
    ! [B4: set_set_nat,X: nat,A4: set_nat] :
      ( ! [C5: set_nat] :
          ( ( member_set_nat @ C5 @ B4 )
         => ~ ( member_nat @ X @ C5 ) )
     => ( ( member_set_nat @ A4 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B4 ) )
        = ( ( member_nat @ X @ A4 )
          & ( member_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_3302_mult__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3303_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3304_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3305_mult__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3306_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3307_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3308_mult__left__less__imp__less,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3309_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3310_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3311_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3312_mult__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3313_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3314_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3315_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3316_mult__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3317_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3318_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3319_mult__right__less__imp__less,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3320_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3321_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3322_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3323_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3324_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3325_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3326_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3327_mult__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3328_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3329_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3330_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3331_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3332_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3333_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3334_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3335_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3336_mult__left__le__imp__le,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3337_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3338_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3339_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3340_mult__right__le__imp__le,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3341_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3342_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3343_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3344_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3345_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3346_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3347_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3348_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3349_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3350_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3351_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3352_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3353_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3354_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3355_mult__left__le,axiom,
    ! [C2: real,A: real] :
      ( ( ord_less_eq_real @ C2 @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_3356_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_3357_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_3358_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_3359_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_3360_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_3361_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_3362_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_3363_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3364_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3365_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3366_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3367_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3368_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3369_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_3370_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_3371_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_3372_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3373_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3374_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3375_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3376_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3377_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3378_less__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3379_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3380_less__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3381_less__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).

% less_add_iff1
thf(fact_3382_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).

% less_add_iff1
thf(fact_3383_less__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).

% less_add_iff1
thf(fact_3384_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_3385_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_3386_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_3387_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_3388_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_3389_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_3390_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_3391_mult__le__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3392_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3393_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3394_mult__le__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3395_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3396_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3397_mult__le__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_eq_real @ C2 @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3398_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3399_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3400_mult__le__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3401_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3402_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3403_mult__less__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3404_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3405_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3406_mult__less__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3407_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3408_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3409_mult__less__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_real @ C2 @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3410_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3411_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3412_mult__less__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3413_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3414_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3415_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V2: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V2 )
           => ( ( ( plus_plus_real @ U @ V2 )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3416_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3417_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3418_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M5: nat,N6: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N6 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N6 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3419_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_3420_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_3421_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_3422_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V2: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V2 )
           => ( ( ( plus_plus_real @ U @ V2 )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3423_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3424_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3425_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_3426_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_3427_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_3428_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_3429_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_3430_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3431_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3432_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3433_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_3434_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_3435_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_3436_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_3437_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_3438_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_3439_star__false__left,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P2 )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_3440_star__false__right,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ P2 @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_3441_snga__same__false,axiom,
    ! [P3: array_VEBT_VEBTi,X: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ P3 @ X ) @ ( snga_assn_VEBT_VEBTi @ P3 @ Y ) )
      = bot_bot_assn ) ).

% snga_same_false
thf(fact_3442_mod__h__bot__iff_I5_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_3443_mod__starE,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B @ H_2 ) ) ) ).

% mod_starE
thf(fact_3444_mod__starD,axiom,
    ! [A4: assn,B4: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A4 @ B4 ) @ H2 )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A4 @ H1 )
          & ( rep_assn @ B4 @ H22 ) ) ) ).

% mod_starD
thf(fact_3445_assn__one__left,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ one_one_assn @ P2 )
      = P2 ) ).

% assn_one_left
thf(fact_3446_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_3447_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_3448_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_3449_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3450_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3451_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3452_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3453_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3454_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3455_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3456_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_3457_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_3458_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3459_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3460_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3461_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3462_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3463_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3464_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C2: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C2 ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3465_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C2: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C2 ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3466_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C2: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C2 ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3467_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C2: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C2 ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3468_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3469_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3470_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3471_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_3472_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_3473_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_3474_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_3475_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_3476_minus__Max__eq__Min,axiom,
    ! [S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( uminus1351360451143612070nteger @ ( lattic4901227151466704046nteger @ S3 ) )
          = ( lattic1063845414844153500nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S3 ) ) ) ) ) ).

% minus_Max_eq_Min
thf(fact_3477_minus__Max__eq__Min,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( uminus_uminus_real @ ( lattic4275903605611617917x_real @ S3 ) )
          = ( lattic3629708407755379051n_real @ ( image_real_real @ uminus_uminus_real @ S3 ) ) ) ) ) ).

% minus_Max_eq_Min
thf(fact_3478_minus__Max__eq__Min,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ( ( uminus_uminus_rat @ ( lattic7630753665789217321ax_rat @ S3 ) )
          = ( lattic8086005427650270231in_rat @ ( image_rat_rat @ uminus_uminus_rat @ S3 ) ) ) ) ) ).

% minus_Max_eq_Min
thf(fact_3479_minus__Max__eq__Min,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( uminus_uminus_int @ ( lattic8263393255366662781ax_int @ S3 ) )
          = ( lattic8718645017227715691in_int @ ( image_int_int @ uminus_uminus_int @ S3 ) ) ) ) ) ).

% minus_Max_eq_Min
thf(fact_3480_minus__Min__eq__Max,axiom,
    ! [S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( uminus1351360451143612070nteger @ ( lattic1063845414844153500nteger @ S3 ) )
          = ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S3 ) ) ) ) ) ).

% minus_Min_eq_Max
thf(fact_3481_minus__Min__eq__Max,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( uminus_uminus_real @ ( lattic3629708407755379051n_real @ S3 ) )
          = ( lattic4275903605611617917x_real @ ( image_real_real @ uminus_uminus_real @ S3 ) ) ) ) ) ).

% minus_Min_eq_Max
thf(fact_3482_minus__Min__eq__Max,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ( ( uminus_uminus_rat @ ( lattic8086005427650270231in_rat @ S3 ) )
          = ( lattic7630753665789217321ax_rat @ ( image_rat_rat @ uminus_uminus_rat @ S3 ) ) ) ) ) ).

% minus_Min_eq_Max
thf(fact_3483_minus__Min__eq__Max,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( uminus_uminus_int @ ( lattic8718645017227715691in_int @ S3 ) )
          = ( lattic8263393255366662781ax_int @ ( image_int_int @ uminus_uminus_int @ S3 ) ) ) ) ) ).

% minus_Min_eq_Max
thf(fact_3484_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_3485_add_Oinverse__inverse,axiom,
    ! [A: uint32] :
      ( ( uminus_uminus_uint32 @ ( uminus_uminus_uint32 @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3486_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3487_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3488_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3489_neg__equal__iff__equal,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( uminus_uminus_uint32 @ A )
        = ( uminus_uminus_uint32 @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_3490_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_3491_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_3492_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_3493_Compl__subset__Compl__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( uminus1532241313380277803et_int @ B4 ) )
      = ( ord_less_eq_set_int @ B4 @ A4 ) ) ).

% Compl_subset_Compl_iff
thf(fact_3494_Compl__anti__mono,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B4 ) @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).

% Compl_anti_mono
thf(fact_3495_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_3496_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_3497_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_3498_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_3499_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_3500_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_3501_neg__equal__0__iff__equal,axiom,
    ! [A: uint32] :
      ( ( ( uminus_uminus_uint32 @ A )
        = zero_zero_uint32 )
      = ( A = zero_zero_uint32 ) ) ).

% neg_equal_0_iff_equal
thf(fact_3502_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_3503_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_3504_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_3505_neg__0__equal__iff__equal,axiom,
    ! [A: uint32] :
      ( ( zero_zero_uint32
        = ( uminus_uminus_uint32 @ A ) )
      = ( zero_zero_uint32 = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3506_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3507_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3508_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3509_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_uint32 @ zero_zero_uint32 )
    = zero_zero_uint32 ) ).

% add.inverse_neutral
thf(fact_3510_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_3511_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_3512_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_3513_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_3514_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_3515_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_3516_compl__le__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) )
      = ( ord_less_eq_assn @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_3517_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
      = ( ord_less_eq_set_int @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_3518_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_3519_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_3520_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_3521_compl__less__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) )
      = ( ord_less_assn @ Y @ X ) ) ).

% compl_less_compl_iff
thf(fact_3522_add__minus__cancel,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ A @ ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_3523_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_3524_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_3525_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_3526_minus__add__cancel,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( plus_plus_uint32 @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_3527_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_3528_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_3529_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_3530_minus__add__distrib,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A @ B ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( uminus_uminus_uint32 @ B ) ) ) ).

% minus_add_distrib
thf(fact_3531_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_3532_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_3533_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_3534_minus__diff__eq,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32 @ ( minus_minus_uint32 @ A @ B ) )
      = ( minus_minus_uint32 @ B @ A ) ) ).

% minus_diff_eq
thf(fact_3535_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_3536_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_3537_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_3538_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_3539_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_3540_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_3541_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_3542_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_3543_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_3544_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_3545_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_3546_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_3547_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_3548_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_3549_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_3550_less__neg__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_3551_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_3552_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_3553_neg__less__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_pos
thf(fact_3554_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_3555_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_3556_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_3557_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_3558_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_3559_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_0_iff_less
thf(fact_3560_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_3561_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_3562_ab__left__minus,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ A )
      = zero_zero_uint32 ) ).

% ab_left_minus
thf(fact_3563_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_3564_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_3565_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_3566_add_Oright__inverse,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ ( uminus_uminus_uint32 @ A ) )
      = zero_zero_uint32 ) ).

% add.right_inverse
thf(fact_3567_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_3568_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_3569_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_3570_diff__0,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ A )
      = ( uminus_uminus_uint32 @ A ) ) ).

% diff_0
thf(fact_3571_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_3572_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_3573_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_3574_verit__minus__simplify_I3_J,axiom,
    ! [B: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ B )
      = ( uminus_uminus_uint32 @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_3575_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_3576_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_3577_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_3578_uminus__add__conv__diff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ B )
      = ( minus_minus_uint32 @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3579_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3580_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3581_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_3582_diff__minus__eq__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ A @ ( uminus_uminus_uint32 @ B ) )
      = ( plus_plus_uint32 @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_3583_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_3584_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_3585_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_3586_subset__Compl__singleton,axiom,
    ! [A4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A4 @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( ~ ( member_VEBT_VEBT @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_3587_subset__Compl__singleton,axiom,
    ! [A4: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_3588_subset__Compl__singleton,axiom,
    ! [A4: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_3589_subset__Compl__singleton,axiom,
    ! [A4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_3590_subset__Compl__singleton,axiom,
    ! [A4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_3591_subset__Compl__singleton,axiom,
    ! [A4: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_3592_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(7)
thf(fact_3593_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% add_neg_numeral_special(7)
thf(fact_3594_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% add_neg_numeral_special(7)
thf(fact_3595_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% add_neg_numeral_special(7)
thf(fact_3596_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(8)
thf(fact_3597_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = zero_zero_real ) ).

% add_neg_numeral_special(8)
thf(fact_3598_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = zero_zero_rat ) ).

% add_neg_numeral_special(8)
thf(fact_3599_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = zero_zero_int ) ).

% add_neg_numeral_special(8)
thf(fact_3600_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% diff_numeral_special(12)
thf(fact_3601_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% diff_numeral_special(12)
thf(fact_3602_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_3603_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% diff_numeral_special(12)
thf(fact_3604_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_3605_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_3606_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P5: assn,Q6: assn] : ( times_times_assn @ Q6 @ P5 ) ) ) ).

% assn_times_comm
thf(fact_3607_assn__times__assoc,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P2 @ Q2 ) @ R2 )
      = ( times_times_assn @ P2 @ ( times_times_assn @ Q2 @ R2 ) ) ) ).

% assn_times_assoc
thf(fact_3608_equation__minus__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A
        = ( uminus_uminus_uint32 @ B ) )
      = ( B
        = ( uminus_uminus_uint32 @ A ) ) ) ).

% equation_minus_iff
thf(fact_3609_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_3610_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_3611_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_3612_minus__equation__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( uminus_uminus_uint32 @ A )
        = B )
      = ( ( uminus_uminus_uint32 @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_3613_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_3614_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_3615_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_3616_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_3617_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_3618_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_3619_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_3620_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_3621_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_3622_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_3623_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_3624_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_3625_compl__le__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ ( uminus_uminus_assn @ Y ) @ X )
     => ( ord_less_eq_assn @ ( uminus_uminus_assn @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_3626_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_3627_compl__le__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ ( uminus_uminus_assn @ X ) )
     => ( ord_less_eq_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).

% compl_le_swap1
thf(fact_3628_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
     => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% compl_le_swap1
thf(fact_3629_compl__mono,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ord_less_eq_assn @ ( uminus_uminus_assn @ Y ) @ ( uminus_uminus_assn @ X ) ) ) ).

% compl_mono
thf(fact_3630_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).

% compl_mono
thf(fact_3631_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_3632_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_3633_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_3634_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_3635_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_3636_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_3637_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3638_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3639_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3640_compl__less__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_assn @ ( uminus_uminus_assn @ Y ) @ X )
     => ( ord_less_assn @ ( uminus_uminus_assn @ X ) @ Y ) ) ).

% compl_less_swap2
thf(fact_3641_compl__less__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_assn @ Y @ ( uminus_uminus_assn @ X ) )
     => ( ord_less_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).

% compl_less_swap1
thf(fact_3642_group__cancel_Oneg1,axiom,
    ! [A4: uint32,K: uint32,A: uint32] :
      ( ( A4
        = ( plus_plus_uint32 @ K @ A ) )
     => ( ( uminus_uminus_uint32 @ A4 )
        = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ K ) @ ( uminus_uminus_uint32 @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3643_group__cancel_Oneg1,axiom,
    ! [A4: real,K: real,A: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3644_group__cancel_Oneg1,axiom,
    ! [A4: rat,K: rat,A: rat] :
      ( ( A4
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3645_group__cancel_Oneg1,axiom,
    ! [A4: int,K: int,A: int] :
      ( ( A4
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_3646_add_Oinverse__distrib__swap,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A @ B ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ B ) @ ( uminus_uminus_uint32 @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3647_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3648_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3649_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3650_minus__diff__commute,axiom,
    ! [B: uint32,A: uint32] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ B ) @ A )
      = ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_3651_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_3652_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_3653_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_3654_neg__eq__iff__add__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( uminus_uminus_uint32 @ A )
        = B )
      = ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3655_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3656_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3657_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3658_eq__neg__iff__add__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A
        = ( uminus_uminus_uint32 @ B ) )
      = ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3659_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3660_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3661_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3662_add_Oinverse__unique,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 )
     => ( ( uminus_uminus_uint32 @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_3663_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_3664_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_3665_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_3666_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ A )
      = zero_zero_uint32 ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3667_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3668_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3669_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3670_add__eq__0__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 )
      = ( B
        = ( uminus_uminus_uint32 @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3671_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3672_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3673_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_3674_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_3675_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_3676_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_3677_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% le_minus_one_simps(2)
thf(fact_3678_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_3679_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% le_minus_one_simps(2)
thf(fact_3680_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(4)
thf(fact_3681_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_3682_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(4)
thf(fact_3683_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(4)
thf(fact_3684_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_3685_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(4)
thf(fact_3686_less__minus__one__simps_I2_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% less_minus_one_simps(2)
thf(fact_3687_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_3688_less__minus__one__simps_I2_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% less_minus_one_simps(2)
thf(fact_3689_group__cancel_Osub2,axiom,
    ! [B4: uint32,K: uint32,B: uint32,A: uint32] :
      ( ( B4
        = ( plus_plus_uint32 @ K @ B ) )
     => ( ( minus_minus_uint32 @ A @ B4 )
        = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ K ) @ ( minus_minus_uint32 @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3690_group__cancel_Osub2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3691_group__cancel_Osub2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3692_group__cancel_Osub2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3693_diff__conv__add__uminus,axiom,
    ( minus_minus_uint32
    = ( ^ [A7: uint32,B7: uint32] : ( plus_plus_uint32 @ A7 @ ( uminus_uminus_uint32 @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3694_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A7: real,B7: real] : ( plus_plus_real @ A7 @ ( uminus_uminus_real @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3695_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A7: rat,B7: rat] : ( plus_plus_rat @ A7 @ ( uminus_uminus_rat @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3696_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A7: int,B7: int] : ( plus_plus_int @ A7 @ ( uminus_uminus_int @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3697_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_uint32
    = ( ^ [A7: uint32,B7: uint32] : ( plus_plus_uint32 @ A7 @ ( uminus_uminus_uint32 @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3698_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A7: real,B7: real] : ( plus_plus_real @ A7 @ ( uminus_uminus_real @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3699_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A7: rat,B7: rat] : ( plus_plus_rat @ A7 @ ( uminus_uminus_rat @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3700_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A7: int,B7: int] : ( plus_plus_int @ A7 @ ( uminus_uminus_int @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3701_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_3702_subset__Compl__self__eq,axiom,
    ! [A4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_3703_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_3704_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% le_minus_one_simps(1)
thf(fact_3705_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_3706_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% le_minus_one_simps(1)
thf(fact_3707_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(3)
thf(fact_3708_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_3709_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(3)
thf(fact_3710_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(3)
thf(fact_3711_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_3712_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(3)
thf(fact_3713_less__minus__one__simps_I1_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% less_minus_one_simps(1)
thf(fact_3714_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_3715_less__minus__one__simps_I1_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% less_minus_one_simps(1)
thf(fact_3716_Compl__insert,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) )
      = ( minus_5127226145743854075T_VEBT @ ( uminus8041839845116263051T_VEBT @ A4 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Compl_insert
thf(fact_3717_Compl__insert,axiom,
    ! [X: int,A4: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A4 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_3718_Compl__insert,axiom,
    ! [X: $o,A4: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X @ A4 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A4 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_3719_Compl__insert,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A4 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_3720_frac__neg,axiom,
    ! [X: real] :
      ( ( ( member_real @ X @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
          = zero_zero_real ) )
      & ( ~ ( member_real @ X @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).

% frac_neg
thf(fact_3721_frac__neg,axiom,
    ! [X: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).

% frac_neg
thf(fact_3722_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu965353292909893953uint32 @ zero_zero_uint32 )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% dbl_dec_simps(2)
thf(fact_3723_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_3724_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_3725_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_3726_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_3727_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_3728_Gcd__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( gcd_Gcd_nat @ A4 )
        = zero_zero_nat )
      = ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% Gcd_0_iff
thf(fact_3729_Gcd__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ( gcd_Gcd_int @ A4 )
        = zero_zero_int )
      = ( ord_less_eq_set_int @ A4 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).

% Gcd_0_iff
thf(fact_3730_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_3731_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,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_3732_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_nat @ bot_bot_set_nat )
    = none_set_nat ) ).

% the_dflt_None_empty
thf(fact_3733_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_int @ bot_bot_set_int )
    = none_set_int ) ).

% the_dflt_None_empty
thf(fact_3734_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_o @ bot_bot_set_o )
    = none_set_o ) ).

% the_dflt_None_empty
thf(fact_3735_ComplI,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ~ ( member_nat @ C2 @ A4 )
     => ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A4 ) ) ) ).

% ComplI
thf(fact_3736_ComplI,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ C2 @ A4 )
     => ( member_VEBT_VEBT @ C2 @ ( uminus8041839845116263051T_VEBT @ A4 ) ) ) ).

% ComplI
thf(fact_3737_ComplI,axiom,
    ! [C2: real,A4: set_real] :
      ( ~ ( member_real @ C2 @ A4 )
     => ( member_real @ C2 @ ( uminus612125837232591019t_real @ A4 ) ) ) ).

% ComplI
thf(fact_3738_ComplI,axiom,
    ! [C2: int,A4: set_int] :
      ( ~ ( member_int @ C2 @ A4 )
     => ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).

% ComplI
thf(fact_3739_ComplI,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ~ ( member_set_nat @ C2 @ A4 )
     => ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A4 ) ) ) ).

% ComplI
thf(fact_3740_Compl__iff,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A4 ) )
      = ( ~ ( member_nat @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_3741_Compl__iff,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( uminus8041839845116263051T_VEBT @ A4 ) )
      = ( ~ ( member_VEBT_VEBT @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_3742_Compl__iff,axiom,
    ! [C2: real,A4: set_real] :
      ( ( member_real @ C2 @ ( uminus612125837232591019t_real @ A4 ) )
      = ( ~ ( member_real @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_3743_Compl__iff,axiom,
    ! [C2: int,A4: set_int] :
      ( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A4 ) )
      = ( ~ ( member_int @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_3744_Compl__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A4 ) )
      = ( ~ ( member_set_nat @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_3745_the__dflt__None__nonempty,axiom,
    ! [S3: set_nat] :
      ( ( S3 != bot_bot_set_nat )
     => ( ( dflt_None_set_nat @ S3 )
        = ( some_set_nat @ S3 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_3746_the__dflt__None__nonempty,axiom,
    ! [S3: set_int] :
      ( ( S3 != bot_bot_set_int )
     => ( ( dflt_None_set_int @ S3 )
        = ( some_set_int @ S3 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_3747_the__dflt__None__nonempty,axiom,
    ! [S3: set_o] :
      ( ( S3 != bot_bot_set_o )
     => ( ( dflt_None_set_o @ S3 )
        = ( some_set_o @ S3 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_3748_Gcd__empty,axiom,
    ( ( gcd_Gcd_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_empty
thf(fact_3749_Gcd__empty,axiom,
    ( ( gcd_Gcd_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_empty
thf(fact_3750_ComplD,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A4 ) )
     => ~ ( member_nat @ C2 @ A4 ) ) ).

% ComplD
thf(fact_3751_ComplD,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( uminus8041839845116263051T_VEBT @ A4 ) )
     => ~ ( member_VEBT_VEBT @ C2 @ A4 ) ) ).

% ComplD
thf(fact_3752_ComplD,axiom,
    ! [C2: real,A4: set_real] :
      ( ( member_real @ C2 @ ( uminus612125837232591019t_real @ A4 ) )
     => ~ ( member_real @ C2 @ A4 ) ) ).

% ComplD
thf(fact_3753_ComplD,axiom,
    ! [C2: int,A4: set_int] :
      ( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A4 ) )
     => ~ ( member_int @ C2 @ A4 ) ) ).

% ComplD
thf(fact_3754_ComplD,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A4 ) )
     => ~ ( member_set_nat @ C2 @ A4 ) ) ).

% ComplD
thf(fact_3755_dflt__None__set__def,axiom,
    ( dflt_None_set_nat
    = ( ^ [S6: set_nat] : ( if_option_set_nat @ ( S6 = bot_bot_set_nat ) @ none_set_nat @ ( some_set_nat @ S6 ) ) ) ) ).

% dflt_None_set_def
thf(fact_3756_dflt__None__set__def,axiom,
    ( dflt_None_set_int
    = ( ^ [S6: set_int] : ( if_option_set_int @ ( S6 = bot_bot_set_int ) @ none_set_int @ ( some_set_int @ S6 ) ) ) ) ).

% dflt_None_set_def
thf(fact_3757_dflt__None__set__def,axiom,
    ( dflt_None_set_o
    = ( ^ [S6: set_o] : ( if_option_set_o @ ( S6 = bot_bot_set_o ) @ none_set_o @ ( some_set_o @ S6 ) ) ) ) ).

% dflt_None_set_def
thf(fact_3758_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,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_3759_the__dflt__None__set,axiom,
    ! [X: set_nat] :
      ( ( the_default_set_nat @ bot_bot_set_nat @ ( dflt_None_set_nat @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_3760_the__dflt__None__set,axiom,
    ! [X: set_int] :
      ( ( the_default_set_int @ bot_bot_set_int @ ( dflt_None_set_int @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_3761_the__dflt__None__set,axiom,
    ! [X: set_o] :
      ( ( the_default_set_o @ bot_bot_set_o @ ( dflt_None_set_o @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_3762_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( X3 = Y ) )
       => ( ( foldr_nat_nat @ plus_plus_nat @ Xs @ zero_zero_nat )
          = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ Y ) ) ) ) ).

% foldr_same_int
thf(fact_3763_Gcd__fin__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( semiri4258706085729940814in_nat @ A4 )
        = zero_zero_nat )
      = ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
        & ( finite_finite_nat @ A4 ) ) ) ).

% Gcd_fin_0_iff
thf(fact_3764_Gcd__fin__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ( semiri4256215615220890538in_int @ A4 )
        = zero_zero_int )
      = ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
        & ( finite_finite_int @ A4 ) ) ) ).

% Gcd_fin_0_iff
thf(fact_3765_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3766_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3767_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3768_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3769_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( insert_VEBT_VEBT @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3770_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_int,X: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3771_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_o,X: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3772_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_complex,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ ( insert_complex @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3773_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_literal,X: literal] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_literal @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_literal @ ( insert_literal @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3774_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_list_nat,X: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3775_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_set_nat,X: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3776_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_3777_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( count_list_int @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_3778_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( count_list_set_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_3779_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( count_list_VEBT_VEBT @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_3780_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( count_list_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_3781_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ( count_list_real @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_3782_count__notin,axiom,
    ! [X: $o,Xs: list_o] :
      ( ~ ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ( count_list_o @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_3783_image__Fpow__mono,axiom,
    ! [F: nat > set_nat,A4: set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A4 ) @ B4 )
     => ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A4 ) ) @ ( finite_Fpow_set_nat @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_3784_image__Fpow__mono,axiom,
    ! [F: nat > nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B4 )
     => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A4 ) ) @ ( finite_Fpow_nat @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_3785_image__Fpow__mono,axiom,
    ! [F: int > nat,A4: set_int,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B4 )
     => ( ord_le6893508408891458716et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ ( finite_Fpow_int @ A4 ) ) @ ( finite_Fpow_nat @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_3786_image__Fpow__mono,axiom,
    ! [F: nat > int,A4: set_nat,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B4 )
     => ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A4 ) ) @ ( finite_Fpow_int @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_3787_image__Fpow__mono,axiom,
    ! [F: int > int,A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B4 )
     => ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( finite_Fpow_int @ A4 ) ) @ ( finite_Fpow_int @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_3788_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_3789_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_3790_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_3791_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_3792_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_3793_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3794_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3795_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3796_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3797_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_uint32 @ zero_zero_uint32 @ ( suc @ N ) )
      = zero_zero_uint32 ) ).

% power_0_Suc
thf(fact_3798_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_3799_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_3800_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_3801_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_3802_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_3803_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3804_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3805_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3806_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3807_card_Oempty,axiom,
    ( ( finite_card_complex @ bot_bot_set_complex )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3808_card_Oempty,axiom,
    ( ( finite_card_literal @ bot_bot_set_literal )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3809_card_Oempty,axiom,
    ( ( finite_card_list_nat @ bot_bot_set_list_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3810_card_Oempty,axiom,
    ( ( finite_card_set_nat @ bot_bot_set_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3811_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3812_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3813_card_Oempty,axiom,
    ( ( finite_card_o @ bot_bot_set_o )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3814_card_Oinfinite,axiom,
    ! [A4: set_literal] :
      ( ~ ( finite5847741373460823677iteral @ A4 )
     => ( ( finite_card_literal @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_3815_card_Oinfinite,axiom,
    ! [A4: set_list_nat] :
      ( ~ ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite_card_list_nat @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_3816_card_Oinfinite,axiom,
    ! [A4: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite_card_set_nat @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_3817_card_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_card_nat @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_3818_card_Oinfinite,axiom,
    ! [A4: set_int] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( finite_card_int @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_3819_card_Oinfinite,axiom,
    ! [A4: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( finite_card_complex @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_3820_card_Oinfinite,axiom,
    ! [A4: set_Code_integer] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ( ( finite4902975817058060853nteger @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_3821_Gcd__fin_Oempty,axiom,
    ( ( semiri4258706085729940814in_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_fin.empty
thf(fact_3822_Gcd__fin_Oempty,axiom,
    ( ( semiri4256215615220890538in_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_fin.empty
thf(fact_3823_Gcd__fin_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( semiri4258706085729940814in_nat @ A4 )
        = one_one_nat ) ) ).

% Gcd_fin.infinite
thf(fact_3824_Gcd__fin_Oinfinite,axiom,
    ! [A4: set_int] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( semiri4256215615220890538in_int @ A4 )
        = one_one_int ) ) ).

% Gcd_fin.infinite
thf(fact_3825_Gcd__fin__eq__Gcd,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( semiri4258706085729940814in_nat @ A4 )
        = ( gcd_Gcd_nat @ A4 ) ) ) ).

% Gcd_fin_eq_Gcd
thf(fact_3826_Gcd__fin__eq__Gcd,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( semiri4256215615220890538in_int @ A4 )
        = ( gcd_Gcd_int @ A4 ) ) ) ).

% Gcd_fin_eq_Gcd
thf(fact_3827_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3828_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3829_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3830_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3831_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( power_power_rat @ A @ N )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3832_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( ( power_power_nat @ A @ N )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3833_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( power_power_real @ A @ N )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3834_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( power_power_complex @ A @ N )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3835_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ( power_power_int @ A @ N )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3836_card__0__eq,axiom,
    ! [A4: set_literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( ( finite_card_literal @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_literal ) ) ) ).

% card_0_eq
thf(fact_3837_card__0__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( ( finite_card_list_nat @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
thf(fact_3838_card__0__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( ( finite_card_set_nat @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
thf(fact_3839_card__0__eq,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( finite_card_complex @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_complex ) ) ) ).

% card_0_eq
thf(fact_3840_card__0__eq,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( finite4902975817058060853nteger @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bo3990330152332043303nteger ) ) ) ).

% card_0_eq
thf(fact_3841_card__0__eq,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( finite_card_nat @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_3842_card__0__eq,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( finite_card_int @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_int ) ) ) ).

% card_0_eq
thf(fact_3843_card__0__eq,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( finite_card_o @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_o ) ) ) ).

% card_0_eq
thf(fact_3844_card__insert__disjoint,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( finite_card_o @ ( insert_o @ X @ A4 ) )
          = ( suc @ ( finite_card_o @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3845_card__insert__disjoint,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ~ ( member_VEBT_VEBT @ X @ A4 )
       => ( ( finite7802652506058667612T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) )
          = ( suc @ ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3846_card__insert__disjoint,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( finite_card_real @ ( insert_real @ X @ A4 ) )
          = ( suc @ ( finite_card_real @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3847_card__insert__disjoint,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ~ ( member_literal @ X @ A4 )
       => ( ( finite_card_literal @ ( insert_literal @ X @ A4 ) )
          = ( suc @ ( finite_card_literal @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3848_card__insert__disjoint,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ~ ( member_list_nat @ X @ A4 )
       => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
          = ( suc @ ( finite_card_list_nat @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3849_card__insert__disjoint,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ~ ( member_set_nat @ X @ A4 )
       => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
          = ( suc @ ( finite_card_set_nat @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3850_card__insert__disjoint,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
          = ( suc @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3851_card__insert__disjoint,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
          = ( suc @ ( finite_card_int @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3852_card__insert__disjoint,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X @ A4 )
       => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
          = ( suc @ ( finite_card_complex @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3853_card__insert__disjoint,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ~ ( member_Code_integer @ X @ A4 )
       => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A4 ) )
          = ( suc @ ( finite4902975817058060853nteger @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_3854_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3855_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3856_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3857_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3858_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3859_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3860_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3861_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3862_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3863_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3864_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3865_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3866_card__Diff__insert,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ~ ( member_o @ A @ B4 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3867_card__Diff__insert,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A4 )
     => ( ~ ( member_VEBT_VEBT @ A @ B4 )
       => ( ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3868_card__Diff__insert,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ~ ( member_real @ A @ B4 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3869_card__Diff__insert,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ~ ( member_int @ A @ B4 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3870_card__Diff__insert,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ A @ A4 )
     => ( ~ ( member_complex @ A @ B4 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3871_card__Diff__insert,axiom,
    ! [A: literal,A4: set_literal,B4: set_literal] :
      ( ( member_literal @ A @ A4 )
     => ( ~ ( member_literal @ A @ B4 )
       => ( ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3872_card__Diff__insert,axiom,
    ! [A: list_nat,A4: set_list_nat,B4: set_list_nat] :
      ( ( member_list_nat @ A @ A4 )
     => ( ~ ( member_list_nat @ A @ B4 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3873_card__Diff__insert,axiom,
    ! [A: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ( ~ ( member_set_nat @ A @ B4 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3874_card__Diff__insert,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ~ ( member_nat @ A @ B4 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_3875_finite__enumerate__mono__iff,axiom,
    ! [S3: set_nat,M: nat,N: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ M @ ( finite_card_nat @ S3 ) )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
         => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M ) @ ( infini8530281810654367211te_nat @ S3 @ N ) )
            = ( ord_less_nat @ M @ N ) ) ) ) ) ).

% finite_enumerate_mono_iff
thf(fact_3876_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_3877_size__neq__size__imp__neq,axiom,
    ! [X: list_real,Y: list_real] :
      ( ( ( size_size_list_real @ X )
       != ( size_size_list_real @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_3878_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_3879_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_3880_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_3881_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% count_le_length
thf(fact_3882_count__le__length,axiom,
    ! [Xs: list_real,X: real] : ( ord_less_eq_nat @ ( count_list_real @ Xs @ X ) @ ( size_size_list_real @ Xs ) ) ).

% count_le_length
thf(fact_3883_count__le__length,axiom,
    ! [Xs: list_o,X: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X ) @ ( size_size_list_o @ Xs ) ) ).

% count_le_length
thf(fact_3884_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).

% count_le_length
thf(fact_3885_card__length,axiom,
    ! [Xs: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ).

% card_length
thf(fact_3886_card__length,axiom,
    ! [Xs: list_literal] : ( ord_less_eq_nat @ ( finite_card_literal @ ( set_literal2 @ Xs ) ) @ ( size_s2501651207091587910iteral @ Xs ) ) ).

% card_length
thf(fact_3887_card__length,axiom,
    ! [Xs: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).

% card_length
thf(fact_3888_card__length,axiom,
    ! [Xs: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ).

% card_length
thf(fact_3889_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% card_length
thf(fact_3890_card__length,axiom,
    ! [Xs: list_real] : ( ord_less_eq_nat @ ( finite_card_real @ ( set_real2 @ Xs ) ) @ ( size_size_list_real @ Xs ) ) ).

% card_length
thf(fact_3891_card__length,axiom,
    ! [Xs: list_o] : ( ord_less_eq_nat @ ( finite_card_o @ ( set_o2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).

% card_length
thf(fact_3892_card__length,axiom,
    ! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% card_length
thf(fact_3893_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N )
       != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_3894_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_3895_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_3896_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_3897_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_3898_length__induct,axiom,
    ! [P2: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_3899_length__induct,axiom,
    ! [P2: list_real > $o,Xs: list_real] :
      ( ! [Xs3: list_real] :
          ( ! [Ys2: list_real] :
              ( ( ord_less_nat @ ( size_size_list_real @ Ys2 ) @ ( size_size_list_real @ Xs3 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_3900_length__induct,axiom,
    ! [P2: list_o > $o,Xs: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_3901_length__induct,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_3902_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N2: nat] :
        ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3903_finite__maxlen,axiom,
    ! [M7: set_list_real] :
      ( ( finite306553202115118035t_real @ M7 )
     => ? [N2: nat] :
        ! [X5: list_real] :
          ( ( member_list_real @ X5 @ M7 )
         => ( ord_less_nat @ ( size_size_list_real @ X5 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3904_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N2: nat] :
        ! [X5: list_o] :
          ( ( member_list_o @ X5 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3905_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N2: nat] :
        ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3906_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_power
thf(fact_3907_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_3908_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_3909_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_power
thf(fact_3910_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3911_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3912_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3913_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3914_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_less_power
thf(fact_3915_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_3916_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_3917_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_less_power
thf(fact_3918_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% one_le_power
thf(fact_3919_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% one_le_power
thf(fact_3920_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_3921_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_3922_card__subset__eq,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ( ord_le7307670543136651348iteral @ A4 @ B4 )
       => ( ( ( finite_card_literal @ A4 )
            = ( finite_card_literal @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_3923_card__subset__eq,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
       => ( ( ( finite_card_list_nat @ A4 )
            = ( finite_card_list_nat @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_3924_card__subset__eq,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
       => ( ( ( finite_card_set_nat @ A4 )
            = ( finite_card_set_nat @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_3925_card__subset__eq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ( ( finite_card_nat @ A4 )
            = ( finite_card_nat @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_3926_card__subset__eq,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( ( finite_card_complex @ A4 )
            = ( finite_card_complex @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_3927_card__subset__eq,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ( ( finite4902975817058060853nteger @ A4 )
            = ( finite4902975817058060853nteger @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_3928_card__subset__eq,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( ( finite_card_int @ A4 )
            = ( finite_card_int @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_3929_infinite__arbitrarily__large,axiom,
    ! [A4: set_literal,N: nat] :
      ( ~ ( finite5847741373460823677iteral @ A4 )
     => ? [B8: set_literal] :
          ( ( finite5847741373460823677iteral @ B8 )
          & ( ( finite_card_literal @ B8 )
            = N )
          & ( ord_le7307670543136651348iteral @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_3930_infinite__arbitrarily__large,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( ~ ( finite8100373058378681591st_nat @ A4 )
     => ? [B8: set_list_nat] :
          ( ( finite8100373058378681591st_nat @ B8 )
          & ( ( finite_card_list_nat @ B8 )
            = N )
          & ( ord_le6045566169113846134st_nat @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_3931_infinite__arbitrarily__large,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( ~ ( finite1152437895449049373et_nat @ A4 )
     => ? [B8: set_set_nat] :
          ( ( finite1152437895449049373et_nat @ B8 )
          & ( ( finite_card_set_nat @ B8 )
            = N )
          & ( ord_le6893508408891458716et_nat @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_3932_infinite__arbitrarily__large,axiom,
    ! [A4: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ? [B8: set_nat] :
          ( ( finite_finite_nat @ B8 )
          & ( ( finite_card_nat @ B8 )
            = N )
          & ( ord_less_eq_set_nat @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_3933_infinite__arbitrarily__large,axiom,
    ! [A4: set_complex,N: nat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ? [B8: set_complex] :
          ( ( finite3207457112153483333omplex @ B8 )
          & ( ( finite_card_complex @ B8 )
            = N )
          & ( ord_le211207098394363844omplex @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_3934_infinite__arbitrarily__large,axiom,
    ! [A4: set_Code_integer,N: nat] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ? [B8: set_Code_integer] :
          ( ( finite6017078050557962740nteger @ B8 )
          & ( ( finite4902975817058060853nteger @ B8 )
            = N )
          & ( ord_le7084787975880047091nteger @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_3935_infinite__arbitrarily__large,axiom,
    ! [A4: set_int,N: nat] :
      ( ~ ( finite_finite_int @ A4 )
     => ? [B8: set_int] :
          ( ( finite_finite_int @ B8 )
          & ( ( finite_card_int @ B8 )
            = N )
          & ( ord_less_eq_set_int @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_3936_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,R: vEBT_VEBT > vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A4 )
           => ? [B9: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: vEBT_VEBT,A22: vEBT_VEBT,B3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A1 @ A4 )
             => ( ( member_VEBT_VEBT @ A22 @ A4 )
               => ( ( member_VEBT_VEBT @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ ( finite7802652506058667612T_VEBT @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3937_card__le__if__inj__on__rel,axiom,
    ! [B4: set_real,A4: set_VEBT_VEBT,R: vEBT_VEBT > real > $o] :
      ( ( finite_finite_real @ B4 )
     => ( ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A4 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: vEBT_VEBT,A22: vEBT_VEBT,B3: real] :
              ( ( member_VEBT_VEBT @ A1 @ A4 )
             => ( ( member_VEBT_VEBT @ A22 @ A4 )
               => ( ( member_real @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ ( finite_card_real @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3938_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_real,R: real > vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ? [B9: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: real,A22: real,B3: vEBT_VEBT] :
              ( ( member_real @ A1 @ A4 )
             => ( ( member_real @ A22 @ A4 )
               => ( ( member_VEBT_VEBT @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ ( finite7802652506058667612T_VEBT @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3939_card__le__if__inj__on__rel,axiom,
    ! [B4: set_real,A4: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ B4 )
     => ( ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: real,A22: real,B3: real] :
              ( ( member_real @ A1 @ A4 )
             => ( ( member_real @ A22 @ A4 )
               => ( ( member_real @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ ( finite_card_real @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3940_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_int,R: int > vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B9: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: int,A22: int,B3: vEBT_VEBT] :
              ( ( member_int @ A1 @ A4 )
             => ( ( member_int @ A22 @ A4 )
               => ( ( member_VEBT_VEBT @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite7802652506058667612T_VEBT @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3941_card__le__if__inj__on__rel,axiom,
    ! [B4: set_real,A4: set_int,R: int > real > $o] :
      ( ( finite_finite_real @ B4 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: int,A22: int,B3: real] :
              ( ( member_int @ A1 @ A4 )
             => ( ( member_int @ A22 @ A4 )
               => ( ( member_real @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_real @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3942_card__le__if__inj__on__rel,axiom,
    ! [B4: set_literal,A4: set_VEBT_VEBT,R: vEBT_VEBT > literal > $o] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A4 )
           => ? [B9: literal] :
                ( ( member_literal @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: vEBT_VEBT,A22: vEBT_VEBT,B3: literal] :
              ( ( member_VEBT_VEBT @ A1 @ A4 )
             => ( ( member_VEBT_VEBT @ A22 @ A4 )
               => ( ( member_literal @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ ( finite_card_literal @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3943_card__le__if__inj__on__rel,axiom,
    ! [B4: set_literal,A4: set_real,R: real > literal > $o] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ? [B9: literal] :
                ( ( member_literal @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: real,A22: real,B3: literal] :
              ( ( member_real @ A1 @ A4 )
             => ( ( member_real @ A22 @ A4 )
               => ( ( member_literal @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ ( finite_card_literal @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3944_card__le__if__inj__on__rel,axiom,
    ! [B4: set_literal,A4: set_int,R: int > literal > $o] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B9: literal] :
                ( ( member_literal @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: int,A22: int,B3: literal] :
              ( ( member_int @ A1 @ A4 )
             => ( ( member_int @ A22 @ A4 )
               => ( ( member_literal @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_literal @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3945_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_complex,R: complex > vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ? [B9: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ B9 @ B4 )
                & ( R @ A3 @ B9 ) ) )
       => ( ! [A1: complex,A22: complex,B3: vEBT_VEBT] :
              ( ( member_complex @ A1 @ A4 )
             => ( ( member_complex @ A22 @ A4 )
               => ( ( member_VEBT_VEBT @ B3 @ B4 )
                 => ( ( R @ A1 @ B3 )
                   => ( ( R @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite7802652506058667612T_VEBT @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_3946_empty__in__Fpow,axiom,
    ! [A4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A4 ) ) ).

% empty_in_Fpow
thf(fact_3947_empty__in__Fpow,axiom,
    ! [A4: set_int] : ( member_set_int @ bot_bot_set_int @ ( finite_Fpow_int @ A4 ) ) ).

% empty_in_Fpow
thf(fact_3948_empty__in__Fpow,axiom,
    ! [A4: set_o] : ( member_set_o @ bot_bot_set_o @ ( finite_Fpow_o @ A4 ) ) ).

% empty_in_Fpow
thf(fact_3949_card__insert__le,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ ( finite7802652506058667612T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3950_card__insert__le,axiom,
    ! [A4: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ ( insert_int @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3951_card__insert__le,axiom,
    ! [A4: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A4 ) @ ( finite_card_o @ ( insert_o @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3952_card__insert__le,axiom,
    ! [A4: set_complex,X: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ ( insert_complex @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3953_card__insert__le,axiom,
    ! [A4: set_literal,X: literal] : ( ord_less_eq_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ ( insert_literal @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3954_card__insert__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3955_card__insert__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3956_card__insert__le,axiom,
    ! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_3957_power__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_Suc
thf(fact_3958_power__Suc,axiom,
    ! [A: assn,N: nat] :
      ( ( power_power_assn @ A @ ( suc @ N ) )
      = ( times_times_assn @ A @ ( power_power_assn @ A @ N ) ) ) ).

% power_Suc
thf(fact_3959_power__Suc,axiom,
    ! [A: uint32,N: nat] :
      ( ( power_power_uint32 @ A @ ( suc @ N ) )
      = ( times_times_uint32 @ A @ ( power_power_uint32 @ A @ N ) ) ) ).

% power_Suc
thf(fact_3960_power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_Suc
thf(fact_3961_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_3962_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_3963_power__Suc2,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3964_power__Suc2,axiom,
    ! [A: assn,N: nat] :
      ( ( power_power_assn @ A @ ( suc @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3965_power__Suc2,axiom,
    ! [A: uint32,N: nat] :
      ( ( power_power_uint32 @ A @ ( suc @ N ) )
      = ( times_times_uint32 @ ( power_power_uint32 @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3966_power__Suc2,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3967_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3968_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3969_Fpow__mono,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ord_le4403425263959731960et_int @ ( finite_Fpow_int @ A4 ) @ ( finite_Fpow_int @ B4 ) ) ) ).

% Fpow_mono
thf(fact_3970_power__0,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ A @ zero_zero_nat )
      = one_one_uint32 ) ).

% power_0
thf(fact_3971_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_3972_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_3973_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_3974_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_3975_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_3976_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_3977_length__removeAll__less__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_3978_length__removeAll__less__eq,axiom,
    ! [X: real,Xs: list_real] : ( ord_less_eq_nat @ ( size_size_list_real @ ( removeAll_real @ X @ Xs ) ) @ ( size_size_list_real @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_3979_length__removeAll__less__eq,axiom,
    ! [X: $o,Xs: list_o] : ( ord_less_eq_nat @ ( size_size_list_o @ ( removeAll_o @ X @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_3980_length__removeAll__less__eq,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_3981_is__singleton__altdef,axiom,
    ( is_singleton_complex
    = ( ^ [A6: set_complex] :
          ( ( finite_card_complex @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_3982_is__singleton__altdef,axiom,
    ( is_singleton_literal
    = ( ^ [A6: set_literal] :
          ( ( finite_card_literal @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_3983_is__singleton__altdef,axiom,
    ( is_sin2641923865335537900st_nat
    = ( ^ [A6: set_list_nat] :
          ( ( finite_card_list_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_3984_is__singleton__altdef,axiom,
    ( is_singleton_set_nat
    = ( ^ [A6: set_set_nat] :
          ( ( finite_card_set_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_3985_is__singleton__altdef,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( ( finite_card_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_3986_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3987_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3988_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3989_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3990_card__ge__0__finite,axiom,
    ! [A4: set_literal] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ A4 ) )
     => ( finite5847741373460823677iteral @ A4 ) ) ).

% card_ge_0_finite
thf(fact_3991_card__ge__0__finite,axiom,
    ! [A4: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A4 ) )
     => ( finite8100373058378681591st_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_3992_card__ge__0__finite,axiom,
    ! [A4: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A4 ) )
     => ( finite1152437895449049373et_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_3993_card__ge__0__finite,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
     => ( finite_finite_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_3994_card__ge__0__finite,axiom,
    ! [A4: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
     => ( finite_finite_int @ A4 ) ) ).

% card_ge_0_finite
thf(fact_3995_card__ge__0__finite,axiom,
    ! [A4: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
     => ( finite3207457112153483333omplex @ A4 ) ) ).

% card_ge_0_finite
thf(fact_3996_card__ge__0__finite,axiom,
    ! [A4: set_Code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A4 ) )
     => ( finite6017078050557962740nteger @ A4 ) ) ).

% card_ge_0_finite
thf(fact_3997_card__eq__0__iff,axiom,
    ! [A4: set_literal] :
      ( ( ( finite_card_literal @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_literal )
        | ~ ( finite5847741373460823677iteral @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_3998_card__eq__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_list_nat )
        | ~ ( finite8100373058378681591st_nat @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_3999_card__eq__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_set_nat )
        | ~ ( finite1152437895449049373et_nat @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_4000_card__eq__0__iff,axiom,
    ! [A4: set_complex] :
      ( ( ( finite_card_complex @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_complex )
        | ~ ( finite3207457112153483333omplex @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_4001_card__eq__0__iff,axiom,
    ! [A4: set_Code_integer] :
      ( ( ( finite4902975817058060853nteger @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bo3990330152332043303nteger )
        | ~ ( finite6017078050557962740nteger @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_4002_card__eq__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( finite_card_nat @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_4003_card__eq__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ( finite_card_int @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_int )
        | ~ ( finite_finite_int @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_4004_card__eq__0__iff,axiom,
    ! [A4: set_o] :
      ( ( ( finite_card_o @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_o )
        | ~ ( finite_finite_o @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_4005_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_4006_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_4007_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_4008_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_4009_card__insert__if,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X @ A4 )
         => ( ( finite_card_o @ ( insert_o @ X @ A4 ) )
            = ( finite_card_o @ A4 ) ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ( finite_card_o @ ( insert_o @ X @ A4 ) )
            = ( suc @ ( finite_card_o @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4010_card__insert__if,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( ( member_VEBT_VEBT @ X @ A4 )
         => ( ( finite7802652506058667612T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) )
            = ( finite7802652506058667612T_VEBT @ A4 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A4 )
         => ( ( finite7802652506058667612T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) )
            = ( suc @ ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4011_card__insert__if,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X @ A4 )
         => ( ( finite_card_real @ ( insert_real @ X @ A4 ) )
            = ( finite_card_real @ A4 ) ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ( finite_card_real @ ( insert_real @ X @ A4 ) )
            = ( suc @ ( finite_card_real @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4012_card__insert__if,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( ( member_literal @ X @ A4 )
         => ( ( finite_card_literal @ ( insert_literal @ X @ A4 ) )
            = ( finite_card_literal @ A4 ) ) )
        & ( ~ ( member_literal @ X @ A4 )
         => ( ( finite_card_literal @ ( insert_literal @ X @ A4 ) )
            = ( suc @ ( finite_card_literal @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4013_card__insert__if,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( ( member_list_nat @ X @ A4 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
            = ( finite_card_list_nat @ A4 ) ) )
        & ( ~ ( member_list_nat @ X @ A4 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
            = ( suc @ ( finite_card_list_nat @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4014_card__insert__if,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( ( member_set_nat @ X @ A4 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
            = ( finite_card_set_nat @ A4 ) ) )
        & ( ~ ( member_set_nat @ X @ A4 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
            = ( suc @ ( finite_card_set_nat @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4015_card__insert__if,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X @ A4 )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
            = ( finite_card_nat @ A4 ) ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
            = ( suc @ ( finite_card_nat @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4016_card__insert__if,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
            = ( finite_card_int @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
            = ( suc @ ( finite_card_int @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4017_card__insert__if,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X @ A4 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
            = ( finite_card_complex @ A4 ) ) )
        & ( ~ ( member_complex @ X @ A4 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
            = ( suc @ ( finite_card_complex @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4018_card__insert__if,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( member_Code_integer @ X @ A4 )
         => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A4 ) )
            = ( finite4902975817058060853nteger @ A4 ) ) )
        & ( ~ ( member_Code_integer @ X @ A4 )
         => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A4 ) )
            = ( suc @ ( finite4902975817058060853nteger @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_4019_card__Suc__eq__finite,axiom,
    ! [A4: set_o,K: nat] :
      ( ( ( finite_card_o @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: $o,B6: set_o] :
            ( ( A4
              = ( insert_o @ B7 @ B6 ) )
            & ~ ( member_o @ B7 @ B6 )
            & ( ( finite_card_o @ B6 )
              = K )
            & ( finite_finite_o @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4020_card__Suc__eq__finite,axiom,
    ! [A4: set_VEBT_VEBT,K: nat] :
      ( ( ( finite7802652506058667612T_VEBT @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: vEBT_VEBT,B6: set_VEBT_VEBT] :
            ( ( A4
              = ( insert_VEBT_VEBT @ B7 @ B6 ) )
            & ~ ( member_VEBT_VEBT @ B7 @ B6 )
            & ( ( finite7802652506058667612T_VEBT @ B6 )
              = K )
            & ( finite5795047828879050333T_VEBT @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4021_card__Suc__eq__finite,axiom,
    ! [A4: set_real,K: nat] :
      ( ( ( finite_card_real @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: real,B6: set_real] :
            ( ( A4
              = ( insert_real @ B7 @ B6 ) )
            & ~ ( member_real @ B7 @ B6 )
            & ( ( finite_card_real @ B6 )
              = K )
            & ( finite_finite_real @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4022_card__Suc__eq__finite,axiom,
    ! [A4: set_literal,K: nat] :
      ( ( ( finite_card_literal @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: literal,B6: set_literal] :
            ( ( A4
              = ( insert_literal @ B7 @ B6 ) )
            & ~ ( member_literal @ B7 @ B6 )
            & ( ( finite_card_literal @ B6 )
              = K )
            & ( finite5847741373460823677iteral @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4023_card__Suc__eq__finite,axiom,
    ! [A4: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: list_nat,B6: set_list_nat] :
            ( ( A4
              = ( insert_list_nat @ B7 @ B6 ) )
            & ~ ( member_list_nat @ B7 @ B6 )
            & ( ( finite_card_list_nat @ B6 )
              = K )
            & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4024_card__Suc__eq__finite,axiom,
    ! [A4: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: set_nat,B6: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ B7 @ B6 ) )
            & ~ ( member_set_nat @ B7 @ B6 )
            & ( ( finite_card_set_nat @ B6 )
              = K )
            & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4025_card__Suc__eq__finite,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: nat,B6: set_nat] :
            ( ( A4
              = ( insert_nat @ B7 @ B6 ) )
            & ~ ( member_nat @ B7 @ B6 )
            & ( ( finite_card_nat @ B6 )
              = K )
            & ( finite_finite_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4026_card__Suc__eq__finite,axiom,
    ! [A4: set_int,K: nat] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: int,B6: set_int] :
            ( ( A4
              = ( insert_int @ B7 @ B6 ) )
            & ~ ( member_int @ B7 @ B6 )
            & ( ( finite_card_int @ B6 )
              = K )
            & ( finite_finite_int @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4027_card__Suc__eq__finite,axiom,
    ! [A4: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: complex,B6: set_complex] :
            ( ( A4
              = ( insert_complex @ B7 @ B6 ) )
            & ~ ( member_complex @ B7 @ B6 )
            & ( ( finite_card_complex @ B6 )
              = K )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4028_card__Suc__eq__finite,axiom,
    ! [A4: set_Code_integer,K: nat] :
      ( ( ( finite4902975817058060853nteger @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: code_integer,B6: set_Code_integer] :
            ( ( A4
              = ( insert_Code_integer @ B7 @ B6 ) )
            & ~ ( member_Code_integer @ B7 @ B6 )
            & ( ( finite4902975817058060853nteger @ B6 )
              = K )
            & ( finite6017078050557962740nteger @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_4029_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4030_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4031_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4032_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4033_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4034_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4035_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4036_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4037_card__image__le,axiom,
    ! [A4: set_literal,F: literal > complex] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_complex @ ( image_5274195009022015549omplex @ F @ A4 ) ) @ ( finite_card_literal @ A4 ) ) ) ).

% card_image_le
thf(fact_4038_card__image__le,axiom,
    ! [A4: set_literal,F: literal > literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_literal @ ( image_8195128725298311301iteral @ F @ A4 ) ) @ ( finite_card_literal @ A4 ) ) ) ).

% card_image_le
thf(fact_4039_card__image__le,axiom,
    ! [A4: set_literal,F: literal > nat] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_literal_nat @ F @ A4 ) ) @ ( finite_card_literal @ A4 ) ) ) ).

% card_image_le
thf(fact_4040_card__image__le,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) ) ) ).

% card_image_le
thf(fact_4041_card__image__le,axiom,
    ! [A4: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_complex @ ( image_nat_complex @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) ) ) ).

% card_image_le
thf(fact_4042_card__image__le,axiom,
    ! [A4: set_nat,F: nat > literal] :
      ( ( finite_finite_nat @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_literal @ ( image_nat_literal @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) ) ) ).

% card_image_le
thf(fact_4043_card__image__le,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) ) ) ).

% card_image_le
thf(fact_4044_card__image__le,axiom,
    ! [A4: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_int @ F @ A4 ) ) @ ( finite_card_int @ A4 ) ) ) ).

% card_image_le
thf(fact_4045_card__image__le,axiom,
    ! [A4: set_int,F: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_complex @ ( image_int_complex @ F @ A4 ) ) @ ( finite_card_int @ A4 ) ) ) ).

% card_image_le
thf(fact_4046_card__image__le,axiom,
    ! [A4: set_int,F: int > literal] :
      ( ( finite_finite_int @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_literal @ ( image_int_literal @ F @ A4 ) ) @ ( finite_card_int @ A4 ) ) ) ).

% card_image_le
thf(fact_4047_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4048_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4049_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4050_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4051_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4052_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4053_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4054_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4055_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_literal,C3: nat] :
      ( ! [G2: set_literal] :
          ( ( ord_le7307670543136651348iteral @ G2 @ F2 )
         => ( ( finite5847741373460823677iteral @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_literal @ G2 ) @ C3 ) ) )
     => ( ( finite5847741373460823677iteral @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_literal @ F2 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_4056_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_list_nat,C3: nat] :
      ( ! [G2: set_list_nat] :
          ( ( ord_le6045566169113846134st_nat @ G2 @ F2 )
         => ( ( finite8100373058378681591st_nat @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_list_nat @ G2 ) @ C3 ) ) )
     => ( ( finite8100373058378681591st_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_4057_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_set_nat,C3: nat] :
      ( ! [G2: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ G2 @ F2 )
         => ( ( finite1152437895449049373et_nat @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ G2 ) @ C3 ) ) )
     => ( ( finite1152437895449049373et_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_4058_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C3: nat] :
      ( ! [G2: set_nat] :
          ( ( ord_less_eq_set_nat @ G2 @ F2 )
         => ( ( finite_finite_nat @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
     => ( ( finite_finite_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_4059_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_complex,C3: nat] :
      ( ! [G2: set_complex] :
          ( ( ord_le211207098394363844omplex @ G2 @ F2 )
         => ( ( finite3207457112153483333omplex @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_complex @ G2 ) @ C3 ) ) )
     => ( ( finite3207457112153483333omplex @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_complex @ F2 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_4060_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Code_integer,C3: nat] :
      ( ! [G2: set_Code_integer] :
          ( ( ord_le7084787975880047091nteger @ G2 @ F2 )
         => ( ( finite6017078050557962740nteger @ G2 )
           => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ G2 ) @ C3 ) ) )
     => ( ( finite6017078050557962740nteger @ F2 )
        & ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ F2 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_4061_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_int,C3: nat] :
      ( ! [G2: set_int] :
          ( ( ord_less_eq_set_int @ G2 @ F2 )
         => ( ( finite_finite_int @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_int @ G2 ) @ C3 ) ) )
     => ( ( finite_finite_int @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_int @ F2 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_4062_card__seteq,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ( ord_le7307670543136651348iteral @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_literal @ B4 ) @ ( finite_card_literal @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_4063_card__seteq,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B4 ) @ ( finite_card_list_nat @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_4064_card__seteq,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B4 ) @ ( finite_card_set_nat @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_4065_card__seteq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_4066_card__seteq,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ B4 ) @ ( finite_card_complex @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_4067_card__seteq,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ B4 ) @ ( finite4902975817058060853nteger @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_4068_card__seteq,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B4 ) @ ( finite_card_int @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_4069_card__mono,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ( ord_le7307670543136651348iteral @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ B4 ) ) ) ) ).

% card_mono
thf(fact_4070_card__mono,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ).

% card_mono
thf(fact_4071_card__mono,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) ) ) ) ).

% card_mono
thf(fact_4072_card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).

% card_mono
thf(fact_4073_card__mono,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B4 ) ) ) ) ).

% card_mono
thf(fact_4074_card__mono,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B4 ) ) ) ) ).

% card_mono
thf(fact_4075_card__mono,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B4 ) ) ) ) ).

% card_mono
thf(fact_4076_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_literal] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_literal @ S3 ) )
     => ~ ! [T4: set_literal] :
            ( ( ord_le7307670543136651348iteral @ T4 @ S3 )
           => ( ( ( finite_card_literal @ T4 )
                = N )
             => ~ ( finite5847741373460823677iteral @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_4077_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_list_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S3 ) )
     => ~ ! [T4: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T4 @ S3 )
           => ( ( ( finite_card_list_nat @ T4 )
                = N )
             => ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_4078_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S3 ) )
     => ~ ! [T4: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ T4 @ S3 )
           => ( ( ( finite_card_set_nat @ T4 )
                = N )
             => ~ ( finite1152437895449049373et_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_4079_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S3 ) )
     => ~ ! [T4: set_nat] :
            ( ( ord_less_eq_set_nat @ T4 @ S3 )
           => ( ( ( finite_card_nat @ T4 )
                = N )
             => ~ ( finite_finite_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_4080_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_complex] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S3 ) )
     => ~ ! [T4: set_complex] :
            ( ( ord_le211207098394363844omplex @ T4 @ S3 )
           => ( ( ( finite_card_complex @ T4 )
                = N )
             => ~ ( finite3207457112153483333omplex @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_4081_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_Code_integer] :
      ( ( ord_less_eq_nat @ N @ ( finite4902975817058060853nteger @ S3 ) )
     => ~ ! [T4: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ T4 @ S3 )
           => ( ( ( finite4902975817058060853nteger @ T4 )
                = N )
             => ~ ( finite6017078050557962740nteger @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_4082_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_int] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_int @ S3 ) )
     => ~ ! [T4: set_int] :
            ( ( ord_less_eq_set_int @ T4 @ S3 )
           => ( ( ( finite_card_int @ T4 )
                = N )
             => ~ ( finite_finite_int @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_4083_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4084_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4085_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4086_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4087_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_uint32 @ zero_zero_uint32 @ N )
          = one_one_uint32 ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_uint32 @ zero_zero_uint32 @ N )
          = zero_zero_uint32 ) ) ) ).

% power_0_left
thf(fact_4088_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_4089_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_4090_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_4091_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_4092_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_4093_card__1__singletonE,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( ( finite7802652506058667612T_VEBT @ A4 )
        = one_one_nat )
     => ~ ! [X3: vEBT_VEBT] :
            ( A4
           != ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% card_1_singletonE
thf(fact_4094_card__1__singletonE,axiom,
    ! [A4: set_complex] :
      ( ( ( finite_card_complex @ A4 )
        = one_one_nat )
     => ~ ! [X3: complex] :
            ( A4
           != ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ).

% card_1_singletonE
thf(fact_4095_card__1__singletonE,axiom,
    ! [A4: set_literal] :
      ( ( ( finite_card_literal @ A4 )
        = one_one_nat )
     => ~ ! [X3: literal] :
            ( A4
           != ( insert_literal @ X3 @ bot_bot_set_literal ) ) ) ).

% card_1_singletonE
thf(fact_4096_card__1__singletonE,axiom,
    ! [A4: set_list_nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = one_one_nat )
     => ~ ! [X3: list_nat] :
            ( A4
           != ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).

% card_1_singletonE
thf(fact_4097_card__1__singletonE,axiom,
    ! [A4: set_set_nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = one_one_nat )
     => ~ ! [X3: set_nat] :
            ( A4
           != ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_4098_card__1__singletonE,axiom,
    ! [A4: set_nat] :
      ( ( ( finite_card_nat @ A4 )
        = one_one_nat )
     => ~ ! [X3: nat] :
            ( A4
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_4099_card__1__singletonE,axiom,
    ! [A4: set_int] :
      ( ( ( finite_card_int @ A4 )
        = one_one_nat )
     => ~ ! [X3: int] :
            ( A4
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% card_1_singletonE
thf(fact_4100_card__1__singletonE,axiom,
    ! [A4: set_o] :
      ( ( ( finite_card_o @ A4 )
        = one_one_nat )
     => ~ ! [X3: $o] :
            ( A4
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% card_1_singletonE
thf(fact_4101_power__strict__increasing,axiom,
    ! [N: nat,N8: nat,A: real] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N8 ) ) ) ) ).

% power_strict_increasing
thf(fact_4102_power__strict__increasing,axiom,
    ! [N: nat,N8: nat,A: rat] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N8 ) ) ) ) ).

% power_strict_increasing
thf(fact_4103_power__strict__increasing,axiom,
    ! [N: nat,N8: nat,A: nat] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N8 ) ) ) ) ).

% power_strict_increasing
thf(fact_4104_power__strict__increasing,axiom,
    ! [N: nat,N8: nat,A: int] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N8 ) ) ) ) ).

% power_strict_increasing
thf(fact_4105_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_4106_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_4107_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_4108_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_4109_card__less__sym__Diff,axiom,
    ! [A4: set_literal,B4: set_literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( finite5847741373460823677iteral @ B4 )
       => ( ( ord_less_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ B4 ) )
         => ( ord_less_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ B4 ) ) @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ B4 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_4110_card__less__sym__Diff,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite8100373058378681591st_nat @ B4 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B4 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_4111_card__less__sym__Diff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite1152437895449049373et_nat @ B4 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B4 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_4112_card__less__sym__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ( ord_less_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B4 ) )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B4 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_4113_card__less__sym__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ord_less_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B4 ) )
         => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_4114_card__less__sym__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ( ord_less_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B4 ) )
         => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ B4 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_4115_card__less__sym__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_4116_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_uint32 @ zero_zero_uint32 @ N )
        = zero_zero_uint32 ) ) ).

% zero_power
thf(fact_4117_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_rat @ zero_zero_rat @ N )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_4118_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_nat @ zero_zero_nat @ N )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_4119_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ zero_zero_real @ N )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_4120_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ zero_zero_complex @ N )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_4121_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_int @ zero_zero_int @ N )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_4122_power__increasing,axiom,
    ! [N: nat,N8: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N8 ) ) ) ) ).

% power_increasing
thf(fact_4123_power__increasing,axiom,
    ! [N: nat,N8: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N8 ) ) ) ) ).

% power_increasing
thf(fact_4124_power__increasing,axiom,
    ! [N: nat,N8: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N8 ) ) ) ) ).

% power_increasing
thf(fact_4125_power__increasing,axiom,
    ! [N: nat,N8: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N8 ) ) ) ) ).

% power_increasing
thf(fact_4126_card__le__sym__Diff,axiom,
    ! [A4: set_literal,B4: set_literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( finite5847741373460823677iteral @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ B4 ) )
         => ( ord_less_eq_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ B4 ) ) @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ B4 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_4127_card__le__sym__Diff,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite8100373058378681591st_nat @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) )
         => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B4 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_4128_card__le__sym__Diff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite1152437895449049373et_nat @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B4 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_4129_card__le__sym__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B4 ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B4 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_4130_card__le__sym__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B4 ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_4131_card__le__sym__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B4 ) )
         => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ B4 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_4132_card__le__sym__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_4133_psubset__card__mono,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ( ord_less_set_literal @ A4 @ B4 )
       => ( ord_less_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_4134_psubset__card__mono,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ( ord_le1190675801316882794st_nat @ A4 @ B4 )
       => ( ord_less_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_4135_psubset__card__mono,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( ord_less_set_set_nat @ A4 @ B4 )
       => ( ord_less_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_4136_psubset__card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_set_nat @ A4 @ B4 )
       => ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_4137_psubset__card__mono,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_set_int @ A4 @ B4 )
       => ( ord_less_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_4138_psubset__card__mono,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_less_set_complex @ A4 @ B4 )
       => ( ord_less_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_4139_psubset__card__mono,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le1307284697595431911nteger @ A4 @ B4 )
       => ( ord_less_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_4140_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4141_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4142_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4143_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4144_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4145_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4146_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_4147_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_4148_finite__enum__ext,axiom,
    ! [X6: set_nat,Y7: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
         => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
            = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
     => ( ( finite_finite_nat @ X6 )
       => ( ( finite_finite_nat @ Y7 )
         => ( ( ( finite_card_nat @ X6 )
              = ( finite_card_nat @ Y7 ) )
           => ( X6 = Y7 ) ) ) ) ) ).

% finite_enum_ext
thf(fact_4149_finite__enumerate__Ex,axiom,
    ! [S3: set_nat,S: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( member_nat @ S @ S3 )
       => ? [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S3 ) )
            & ( ( infini8530281810654367211te_nat @ S3 @ N2 )
              = S ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_4150_finite__enumerate__in__set,axiom,
    ! [S3: set_nat,N: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
       => ( member_nat @ ( infini8530281810654367211te_nat @ S3 @ N ) @ S3 ) ) ) ).

% finite_enumerate_in_set
thf(fact_4151_length__removeAll__less,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_int @ ( removeAll_int @ X @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_4152_length__removeAll__less,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( removeAll_set_nat @ X @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_4153_length__removeAll__less,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_4154_length__removeAll__less,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_real @ ( removeAll_real @ X @ Xs ) ) @ ( size_size_list_real @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_4155_length__removeAll__less,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_o @ ( removeAll_o @ X @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_4156_length__removeAll__less,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_4157_card__gt__0__iff,axiom,
    ! [A4: set_literal] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ A4 ) )
      = ( ( A4 != bot_bot_set_literal )
        & ( finite5847741373460823677iteral @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4158_card__gt__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A4 ) )
      = ( ( A4 != bot_bot_set_list_nat )
        & ( finite8100373058378681591st_nat @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4159_card__gt__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A4 ) )
      = ( ( A4 != bot_bot_set_set_nat )
        & ( finite1152437895449049373et_nat @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4160_card__gt__0__iff,axiom,
    ! [A4: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
      = ( ( A4 != bot_bot_set_complex )
        & ( finite3207457112153483333omplex @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4161_card__gt__0__iff,axiom,
    ! [A4: set_Code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A4 ) )
      = ( ( A4 != bot_bo3990330152332043303nteger )
        & ( finite6017078050557962740nteger @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4162_card__gt__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
      = ( ( A4 != bot_bot_set_nat )
        & ( finite_finite_nat @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4163_card__gt__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
      = ( ( A4 != bot_bot_set_int )
        & ( finite_finite_int @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4164_card__gt__0__iff,axiom,
    ! [A4: set_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A4 ) )
      = ( ( A4 != bot_bot_set_o )
        & ( finite_finite_o @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_4165_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_literal @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: literal] :
              ( ( member_literal @ X4 @ A4 )
             => ! [Y4: literal] :
                  ( ( member_literal @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_4166_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A4 )
             => ! [Y4: list_nat] :
                  ( ( member_list_nat @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_4167_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A4 )
             => ! [Y4: set_nat] :
                  ( ( member_set_nat @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_4168_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_4169_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ A4 )
             => ! [Y4: int] :
                  ( ( member_int @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_4170_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
             => ! [Y4: complex] :
                  ( ( member_complex @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_4171_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A4 )
             => ! [Y4: code_integer] :
                  ( ( member_Code_integer @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_4172_card__Suc__eq,axiom,
    ! [A4: set_VEBT_VEBT,K: nat] :
      ( ( ( finite7802652506058667612T_VEBT @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: vEBT_VEBT,B6: set_VEBT_VEBT] :
            ( ( A4
              = ( insert_VEBT_VEBT @ B7 @ B6 ) )
            & ~ ( member_VEBT_VEBT @ B7 @ B6 )
            & ( ( finite7802652506058667612T_VEBT @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4173_card__Suc__eq,axiom,
    ! [A4: set_real,K: nat] :
      ( ( ( finite_card_real @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: real,B6: set_real] :
            ( ( A4
              = ( insert_real @ B7 @ B6 ) )
            & ~ ( member_real @ B7 @ B6 )
            & ( ( finite_card_real @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_real ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4174_card__Suc__eq,axiom,
    ! [A4: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: complex,B6: set_complex] :
            ( ( A4
              = ( insert_complex @ B7 @ B6 ) )
            & ~ ( member_complex @ B7 @ B6 )
            & ( ( finite_card_complex @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_complex ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4175_card__Suc__eq,axiom,
    ! [A4: set_literal,K: nat] :
      ( ( ( finite_card_literal @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: literal,B6: set_literal] :
            ( ( A4
              = ( insert_literal @ B7 @ B6 ) )
            & ~ ( member_literal @ B7 @ B6 )
            & ( ( finite_card_literal @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_literal ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4176_card__Suc__eq,axiom,
    ! [A4: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: list_nat,B6: set_list_nat] :
            ( ( A4
              = ( insert_list_nat @ B7 @ B6 ) )
            & ~ ( member_list_nat @ B7 @ B6 )
            & ( ( finite_card_list_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_list_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4177_card__Suc__eq,axiom,
    ! [A4: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: set_nat,B6: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ B7 @ B6 ) )
            & ~ ( member_set_nat @ B7 @ B6 )
            & ( ( finite_card_set_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4178_card__Suc__eq,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: nat,B6: set_nat] :
            ( ( A4
              = ( insert_nat @ B7 @ B6 ) )
            & ~ ( member_nat @ B7 @ B6 )
            & ( ( finite_card_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4179_card__Suc__eq,axiom,
    ! [A4: set_int,K: nat] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: int,B6: set_int] :
            ( ( A4
              = ( insert_int @ B7 @ B6 ) )
            & ~ ( member_int @ B7 @ B6 )
            & ( ( finite_card_int @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_int ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4180_card__Suc__eq,axiom,
    ! [A4: set_o,K: nat] :
      ( ( ( finite_card_o @ A4 )
        = ( suc @ K ) )
      = ( ? [B7: $o,B6: set_o] :
            ( ( A4
              = ( insert_o @ B7 @ B6 ) )
            & ~ ( member_o @ B7 @ B6 )
            & ( ( finite_card_o @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_o ) ) ) ) ) ).

% card_Suc_eq
thf(fact_4181_card__eq__SucD,axiom,
    ! [A4: set_VEBT_VEBT,K: nat] :
      ( ( ( finite7802652506058667612T_VEBT @ A4 )
        = ( suc @ K ) )
     => ? [B3: vEBT_VEBT,B8: set_VEBT_VEBT] :
          ( ( A4
            = ( insert_VEBT_VEBT @ B3 @ B8 ) )
          & ~ ( member_VEBT_VEBT @ B3 @ B8 )
          & ( ( finite7802652506058667612T_VEBT @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% card_eq_SucD
thf(fact_4182_card__eq__SucD,axiom,
    ! [A4: set_real,K: nat] :
      ( ( ( finite_card_real @ A4 )
        = ( suc @ K ) )
     => ? [B3: real,B8: set_real] :
          ( ( A4
            = ( insert_real @ B3 @ B8 ) )
          & ~ ( member_real @ B3 @ B8 )
          & ( ( finite_card_real @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
thf(fact_4183_card__eq__SucD,axiom,
    ! [A4: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ K ) )
     => ? [B3: complex,B8: set_complex] :
          ( ( A4
            = ( insert_complex @ B3 @ B8 ) )
          & ~ ( member_complex @ B3 @ B8 )
          & ( ( finite_card_complex @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
thf(fact_4184_card__eq__SucD,axiom,
    ! [A4: set_literal,K: nat] :
      ( ( ( finite_card_literal @ A4 )
        = ( suc @ K ) )
     => ? [B3: literal,B8: set_literal] :
          ( ( A4
            = ( insert_literal @ B3 @ B8 ) )
          & ~ ( member_literal @ B3 @ B8 )
          & ( ( finite_card_literal @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_literal ) ) ) ) ).

% card_eq_SucD
thf(fact_4185_card__eq__SucD,axiom,
    ! [A4: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ K ) )
     => ? [B3: list_nat,B8: set_list_nat] :
          ( ( A4
            = ( insert_list_nat @ B3 @ B8 ) )
          & ~ ( member_list_nat @ B3 @ B8 )
          & ( ( finite_card_list_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_4186_card__eq__SucD,axiom,
    ! [A4: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ K ) )
     => ? [B3: set_nat,B8: set_set_nat] :
          ( ( A4
            = ( insert_set_nat @ B3 @ B8 ) )
          & ~ ( member_set_nat @ B3 @ B8 )
          & ( ( finite_card_set_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_4187_card__eq__SucD,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ K ) )
     => ? [B3: nat,B8: set_nat] :
          ( ( A4
            = ( insert_nat @ B3 @ B8 ) )
          & ~ ( member_nat @ B3 @ B8 )
          & ( ( finite_card_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_4188_card__eq__SucD,axiom,
    ! [A4: set_int,K: nat] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ K ) )
     => ? [B3: int,B8: set_int] :
          ( ( A4
            = ( insert_int @ B3 @ B8 ) )
          & ~ ( member_int @ B3 @ B8 )
          & ( ( finite_card_int @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
thf(fact_4189_card__eq__SucD,axiom,
    ! [A4: set_o,K: nat] :
      ( ( ( finite_card_o @ A4 )
        = ( suc @ K ) )
     => ? [B3: $o,B8: set_o] :
          ( ( A4
            = ( insert_o @ B3 @ B8 ) )
          & ~ ( member_o @ B3 @ B8 )
          & ( ( finite_card_o @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_o ) ) ) ) ).

% card_eq_SucD
thf(fact_4190_card__1__singleton__iff,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( ( finite7802652506058667612T_VEBT @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: vEBT_VEBT] :
            ( A4
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4191_card__1__singleton__iff,axiom,
    ! [A4: set_complex] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: complex] :
            ( A4
            = ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4192_card__1__singleton__iff,axiom,
    ! [A4: set_literal] :
      ( ( ( finite_card_literal @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: literal] :
            ( A4
            = ( insert_literal @ X4 @ bot_bot_set_literal ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4193_card__1__singleton__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: list_nat] :
            ( A4
            = ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4194_card__1__singleton__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: set_nat] :
            ( A4
            = ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4195_card__1__singleton__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: nat] :
            ( A4
            = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4196_card__1__singleton__iff,axiom,
    ! [A4: set_int] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: int] :
            ( A4
            = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4197_card__1__singleton__iff,axiom,
    ! [A4: set_o] :
      ( ( ( finite_card_o @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: $o] :
            ( A4
            = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% card_1_singleton_iff
thf(fact_4198_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4199_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4200_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4201_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4202_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4203_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4204_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4205_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4206_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_o @ A4 ) )
      = ( ? [A7: $o,B6: set_o] :
            ( ( A4
              = ( insert_o @ A7 @ B6 ) )
            & ~ ( member_o @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_o @ B6 ) )
            & ( finite_finite_o @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4207_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite7802652506058667612T_VEBT @ A4 ) )
      = ( ? [A7: vEBT_VEBT,B6: set_VEBT_VEBT] :
            ( ( A4
              = ( insert_VEBT_VEBT @ A7 @ B6 ) )
            & ~ ( member_VEBT_VEBT @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite7802652506058667612T_VEBT @ B6 ) )
            & ( finite5795047828879050333T_VEBT @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4208_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_real] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A4 ) )
      = ( ? [A7: real,B6: set_real] :
            ( ( A4
              = ( insert_real @ A7 @ B6 ) )
            & ~ ( member_real @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_real @ B6 ) )
            & ( finite_finite_real @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4209_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_literal] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_literal @ A4 ) )
      = ( ? [A7: literal,B6: set_literal] :
            ( ( A4
              = ( insert_literal @ A7 @ B6 ) )
            & ~ ( member_literal @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_literal @ B6 ) )
            & ( finite5847741373460823677iteral @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4210_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_nat @ A4 ) )
      = ( ? [A7: list_nat,B6: set_list_nat] :
            ( ( A4
              = ( insert_list_nat @ A7 @ B6 ) )
            & ~ ( member_list_nat @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ B6 ) )
            & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4211_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_nat @ A4 ) )
      = ( ? [A7: set_nat,B6: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ A7 @ B6 ) )
            & ~ ( member_set_nat @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ B6 ) )
            & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4212_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A4 ) )
      = ( ? [A7: nat,B6: set_nat] :
            ( ( A4
              = ( insert_nat @ A7 @ B6 ) )
            & ~ ( member_nat @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
            & ( finite_finite_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4213_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_int @ A4 ) )
      = ( ? [A7: int,B6: set_int] :
            ( ( A4
              = ( insert_int @ A7 @ B6 ) )
            & ~ ( member_int @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_int @ B6 ) )
            & ( finite_finite_int @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4214_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_complex] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_complex @ A4 ) )
      = ( ? [A7: complex,B6: set_complex] :
            ( ( A4
              = ( insert_complex @ A7 @ B6 ) )
            & ~ ( member_complex @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_complex @ B6 ) )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4215_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_Code_integer] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite4902975817058060853nteger @ A4 ) )
      = ( ? [A7: code_integer,B6: set_Code_integer] :
            ( ( A4
              = ( insert_Code_integer @ A7 @ B6 ) )
            & ~ ( member_Code_integer @ A7 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite4902975817058060853nteger @ B6 ) )
            & ( finite6017078050557962740nteger @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_4216_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_4217_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_4218_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_4219_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_4220_surj__card__le,axiom,
    ! [A4: set_literal,B4: set_complex,F: literal > complex] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_5274195009022015549omplex @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ B4 ) @ ( finite_card_literal @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4221_surj__card__le,axiom,
    ! [A4: set_literal,B4: set_literal,F: literal > literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( ord_le7307670543136651348iteral @ B4 @ ( image_8195128725298311301iteral @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_literal @ B4 ) @ ( finite_card_literal @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4222_surj__card__le,axiom,
    ! [A4: set_literal,B4: set_nat,F: literal > nat] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_literal_nat @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_literal @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4223_surj__card__le,axiom,
    ! [A4: set_nat,B4: set_complex,F: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_nat_complex @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ B4 ) @ ( finite_card_nat @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4224_surj__card__le,axiom,
    ! [A4: set_nat,B4: set_literal,F: nat > literal] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_le7307670543136651348iteral @ B4 @ ( image_nat_literal @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_literal @ B4 ) @ ( finite_card_nat @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4225_surj__card__le,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4226_surj__card__le,axiom,
    ! [A4: set_int,B4: set_complex,F: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_int_complex @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ B4 ) @ ( finite_card_int @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4227_surj__card__le,axiom,
    ! [A4: set_int,B4: set_literal,F: int > literal] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_le7307670543136651348iteral @ B4 @ ( image_int_literal @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_literal @ B4 ) @ ( finite_card_int @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4228_surj__card__le,axiom,
    ! [A4: set_int,B4: set_nat,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_int @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4229_surj__card__le,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_1468599708987790691omplex @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ B4 ) @ ( finite_card_complex @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_4230_power__strict__decreasing,axiom,
    ! [N: nat,N8: nat,A: real] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N8 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4231_power__strict__decreasing,axiom,
    ! [N: nat,N8: nat,A: rat] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N8 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4232_power__strict__decreasing,axiom,
    ! [N: nat,N8: nat,A: nat] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N8 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4233_power__strict__decreasing,axiom,
    ! [N: nat,N8: nat,A: int] :
      ( ( ord_less_nat @ N @ N8 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N8 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_4234_card__1__singletonI,axiom,
    ! [S3: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( finite7802652506058667612T_VEBT @ S3 )
          = one_one_nat )
       => ( ( member_VEBT_VEBT @ X @ S3 )
         => ( S3
            = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4235_card__1__singletonI,axiom,
    ! [S3: set_real,X: real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( finite_card_real @ S3 )
          = one_one_nat )
       => ( ( member_real @ X @ S3 )
         => ( S3
            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4236_card__1__singletonI,axiom,
    ! [S3: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ S3 )
     => ( ( ( finite_card_literal @ S3 )
          = one_one_nat )
       => ( ( member_literal @ X @ S3 )
         => ( S3
            = ( insert_literal @ X @ bot_bot_set_literal ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4237_card__1__singletonI,axiom,
    ! [S3: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ S3 )
     => ( ( ( finite_card_list_nat @ S3 )
          = one_one_nat )
       => ( ( member_list_nat @ X @ S3 )
         => ( S3
            = ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4238_card__1__singletonI,axiom,
    ! [S3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ S3 )
     => ( ( ( finite_card_set_nat @ S3 )
          = one_one_nat )
       => ( ( member_set_nat @ X @ S3 )
         => ( S3
            = ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4239_card__1__singletonI,axiom,
    ! [S3: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( finite_card_complex @ S3 )
          = one_one_nat )
       => ( ( member_complex @ X @ S3 )
         => ( S3
            = ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4240_card__1__singletonI,axiom,
    ! [S3: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( finite4902975817058060853nteger @ S3 )
          = one_one_nat )
       => ( ( member_Code_integer @ X @ S3 )
         => ( S3
            = ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4241_card__1__singletonI,axiom,
    ! [S3: set_nat,X: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( finite_card_nat @ S3 )
          = one_one_nat )
       => ( ( member_nat @ X @ S3 )
         => ( S3
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4242_card__1__singletonI,axiom,
    ! [S3: set_int,X: int] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( finite_card_int @ S3 )
          = one_one_nat )
       => ( ( member_int @ X @ S3 )
         => ( S3
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4243_card__1__singletonI,axiom,
    ! [S3: set_o,X: $o] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( finite_card_o @ S3 )
          = one_one_nat )
       => ( ( member_o @ X @ S3 )
         => ( S3
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% card_1_singletonI
thf(fact_4244_power__decreasing,axiom,
    ! [N: nat,N8: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N8 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4245_power__decreasing,axiom,
    ! [N: nat,N8: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N8 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4246_power__decreasing,axiom,
    ! [N: nat,N8: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N8 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4247_power__decreasing,axiom,
    ! [N: nat,N8: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N8 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N8 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_4248_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4249_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4250_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4251_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_4252_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4253_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4254_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4255_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_4256_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_4257_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_4258_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_4259_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_4260_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_4261_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_4262_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_4263_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_4264_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_4265_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_4266_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_4267_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_4268_card__Diff1__le,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) @ ( finite7802652506058667612T_VEBT @ A4 ) ) ).

% card_Diff1_le
thf(fact_4269_card__Diff1__le,axiom,
    ! [A4: set_complex,X: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A4 ) ) ).

% card_Diff1_le
thf(fact_4270_card__Diff1__le,axiom,
    ! [A4: set_literal,X: literal] : ( ord_less_eq_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) ) @ ( finite_card_literal @ A4 ) ) ).

% card_Diff1_le
thf(fact_4271_card__Diff1__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A4 ) ) ).

% card_Diff1_le
thf(fact_4272_card__Diff1__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A4 ) ) ).

% card_Diff1_le
thf(fact_4273_card__Diff1__le,axiom,
    ! [A4: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A4 ) ) ).

% card_Diff1_le
thf(fact_4274_card__Diff1__le,axiom,
    ! [A4: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A4 ) ) ).

% card_Diff1_le
thf(fact_4275_card__Diff1__le,axiom,
    ! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ).

% card_Diff1_le
thf(fact_4276_card__Diff__subset,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ( ord_le7307670543136651348iteral @ B4 @ A4 )
       => ( ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ B4 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_4277_card__Diff__subset,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ( ord_le6045566169113846134st_nat @ B4 @ A4 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_4278_card__Diff__subset,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_4279_card__Diff__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B4 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_4280_card__Diff__subset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
       => ( ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B4 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_4281_card__Diff__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_4282_card__Diff__subset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ A4 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B4 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_4283_diff__card__le__card__Diff,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ B4 ) ) @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ B4 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_4284_diff__card__le__card__Diff,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_4285_diff__card__le__card__Diff,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B4 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_4286_diff__card__le__card__Diff,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B4 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_4287_diff__card__le__card__Diff,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B4 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_4288_diff__card__le__card__Diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B4 ) ) @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_4289_diff__card__le__card__Diff,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_4290_card__psubset,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( ( finite5847741373460823677iteral @ B4 )
     => ( ( ord_le7307670543136651348iteral @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite_card_literal @ A4 ) @ ( finite_card_literal @ B4 ) )
         => ( ord_less_set_literal @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_4291_card__psubset,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) )
         => ( ord_le1190675801316882794st_nat @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_4292_card__psubset,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) )
         => ( ord_less_set_set_nat @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_4293_card__psubset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
         => ( ord_less_set_nat @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_4294_card__psubset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B4 ) )
         => ( ord_less_set_complex @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_4295_card__psubset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B4 ) )
         => ( ord_le1307284697595431911nteger @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_4296_card__psubset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B4 ) )
         => ( ord_less_set_int @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_4297_finite__enumerate__mono,axiom,
    ! [M: nat,N: nat,S3: set_nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( finite_finite_nat @ S3 )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
         => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M ) @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ) ) ).

% finite_enumerate_mono
thf(fact_4298_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ B @ N ) ) @ ( power_power_nat @ B @ M ) )
        = ( ord_less_nat @ A @ ( power_power_nat @ B @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_4299_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_4300_finite__le__enumerate,axiom,
    ! [S3: set_nat,N: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_4301_card__insert__disjoint_H,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_o @ ( insert_o @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_o @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4302_card__insert__disjoint_H,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ~ ( member_VEBT_VEBT @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite7802652506058667612T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4303_card__insert__disjoint_H,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_real @ ( insert_real @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_real @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4304_card__insert__disjoint_H,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ~ ( member_literal @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_literal @ ( insert_literal @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_literal @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4305_card__insert__disjoint_H,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ~ ( member_list_nat @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_list_nat @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4306_card__insert__disjoint_H,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ~ ( member_set_nat @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_set_nat @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4307_card__insert__disjoint_H,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_nat @ ( insert_nat @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_nat @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4308_card__insert__disjoint_H,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_int @ ( insert_int @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_int @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4309_card__insert__disjoint_H,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite_card_complex @ ( insert_complex @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_complex @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4310_card__insert__disjoint_H,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ~ ( member_Code_integer @ X @ A4 )
       => ( ( minus_minus_nat @ ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A4 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite4902975817058060853nteger @ A4 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_4311_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4312_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4313_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4314_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_4315_card_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ( finite7802652506058667612T_VEBT @ A4 )
          = ( suc @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4316_card_Oremove,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( finite_card_real @ A4 )
          = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4317_card_Oremove,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( member_literal @ X @ A4 )
       => ( ( finite_card_literal @ A4 )
          = ( suc @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4318_card_Oremove,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( member_list_nat @ X @ A4 )
       => ( ( finite_card_list_nat @ A4 )
          = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4319_card_Oremove,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( ( finite_card_set_nat @ A4 )
          = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4320_card_Oremove,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( finite_card_complex @ A4 )
          = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4321_card_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( finite4902975817058060853nteger @ A4 )
          = ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4322_card_Oremove,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( finite_card_int @ A4 )
          = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4323_card_Oremove,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ( finite_card_o @ A4 )
          = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4324_card_Oremove,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( finite_card_nat @ A4 )
          = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_4325_card_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite7802652506058667612T_VEBT @ ( insert_VEBT_VEBT @ X @ A4 ) )
        = ( suc @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4326_card_Oinsert__remove,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( finite_card_literal @ ( insert_literal @ X @ A4 ) )
        = ( suc @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4327_card_Oinsert__remove,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
        = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4328_card_Oinsert__remove,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
        = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4329_card_Oinsert__remove,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
        = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4330_card_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A4 ) )
        = ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4331_card_Oinsert__remove,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
        = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4332_card_Oinsert__remove,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_card_o @ ( insert_o @ X @ A4 ) )
        = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4333_card_Oinsert__remove,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
        = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_4334_card__Suc__Diff1,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ( suc @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) )
          = ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4335_card__Suc__Diff1,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) )
          = ( finite_card_real @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4336_card__Suc__Diff1,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( member_literal @ X @ A4 )
       => ( ( suc @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) ) )
          = ( finite_card_literal @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4337_card__Suc__Diff1,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( member_list_nat @ X @ A4 )
       => ( ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) )
          = ( finite_card_list_nat @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4338_card__Suc__Diff1,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) )
          = ( finite_card_set_nat @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4339_card__Suc__Diff1,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) )
          = ( finite_card_complex @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4340_card__Suc__Diff1,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) )
          = ( finite4902975817058060853nteger @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4341_card__Suc__Diff1,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
          = ( finite_card_int @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4342_card__Suc__Diff1,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
          = ( finite_card_o @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4343_card__Suc__Diff1,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
          = ( finite_card_nat @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_4344_card__Diff1__less,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ord_less_nat @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) @ ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4345_card__Diff1__less,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4346_card__Diff1__less,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( member_literal @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) ) @ ( finite_card_literal @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4347_card__Diff1__less,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( member_list_nat @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4348_card__Diff1__less,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4349_card__Diff1__less,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4350_card__Diff1__less,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4351_card__Diff1__less,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4352_card__Diff1__less,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4353_card__Diff1__less,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_4354_card__Diff2__less,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ( member_VEBT_VEBT @ Y @ A4 )
         => ( ord_less_nat @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) ) ) @ ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4355_card__Diff2__less,axiom,
    ! [A4: set_real,X: real,Y: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( member_real @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) @ ( insert_real @ Y @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4356_card__Diff2__less,axiom,
    ! [A4: set_literal,X: literal,Y: literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( member_literal @ X @ A4 )
       => ( ( member_literal @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) @ ( insert_literal @ Y @ bot_bot_set_literal ) ) ) @ ( finite_card_literal @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4357_card__Diff2__less,axiom,
    ! [A4: set_list_nat,X: list_nat,Y: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( member_list_nat @ X @ A4 )
       => ( ( member_list_nat @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) @ ( insert_list_nat @ Y @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4358_card__Diff2__less,axiom,
    ! [A4: set_set_nat,X: set_nat,Y: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( ( member_set_nat @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4359_card__Diff2__less,axiom,
    ! [A4: set_complex,X: complex,Y: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( member_complex @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ ( insert_complex @ Y @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4360_card__Diff2__less,axiom,
    ! [A4: set_Code_integer,X: code_integer,Y: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( member_Code_integer @ Y @ A4 )
         => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) @ ( insert_Code_integer @ Y @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4361_card__Diff2__less,axiom,
    ! [A4: set_int,X: int,Y: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( member_int @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4362_card__Diff2__less,axiom,
    ! [A4: set_o,X: $o,Y: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ( member_o @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( insert_o @ Y @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4363_card__Diff2__less,axiom,
    ! [A4: set_nat,X: nat,Y: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_4364_card__Diff1__less__iff,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) @ ( finite7802652506058667612T_VEBT @ A4 ) )
      = ( ( finite5795047828879050333T_VEBT @ A4 )
        & ( member_VEBT_VEBT @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4365_card__Diff1__less__iff,axiom,
    ! [A4: set_real,X: real] :
      ( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A4 ) )
      = ( ( finite_finite_real @ A4 )
        & ( member_real @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4366_card__Diff1__less__iff,axiom,
    ! [A4: set_literal,X: literal] :
      ( ( ord_less_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) ) @ ( finite_card_literal @ A4 ) )
      = ( ( finite5847741373460823677iteral @ A4 )
        & ( member_literal @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4367_card__Diff1__less__iff,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A4 ) )
      = ( ( finite8100373058378681591st_nat @ A4 )
        & ( member_list_nat @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4368_card__Diff1__less__iff,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A4 ) )
      = ( ( finite1152437895449049373et_nat @ A4 )
        & ( member_set_nat @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4369_card__Diff1__less__iff,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A4 ) )
      = ( ( finite3207457112153483333omplex @ A4 )
        & ( member_complex @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4370_card__Diff1__less__iff,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A4 ) )
      = ( ( finite6017078050557962740nteger @ A4 )
        & ( member_Code_integer @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4371_card__Diff1__less__iff,axiom,
    ! [A4: set_int,X: int] :
      ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A4 ) )
      = ( ( finite_finite_int @ A4 )
        & ( member_int @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4372_card__Diff1__less__iff,axiom,
    ! [A4: set_o,X: $o] :
      ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A4 ) )
      = ( ( finite_finite_o @ A4 )
        & ( member_o @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4373_card__Diff1__less__iff,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) )
      = ( ( finite_finite_nat @ A4 )
        & ( member_nat @ X @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_4374_card__Diff__singleton__if,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( minus_minus_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_VEBT_VEBT @ X @ A4 )
       => ( ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4375_card__Diff__singleton__if,axiom,
    ! [X: real,A4: set_real] :
      ( ( ( member_real @ X @ A4 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( finite_card_real @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_real @ X @ A4 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) )
          = ( finite_card_real @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4376_card__Diff__singleton__if,axiom,
    ! [X: complex,A4: set_complex] :
      ( ( ( member_complex @ X @ A4 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
          = ( minus_minus_nat @ ( finite_card_complex @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_complex @ X @ A4 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
          = ( finite_card_complex @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4377_card__Diff__singleton__if,axiom,
    ! [X: literal,A4: set_literal] :
      ( ( ( member_literal @ X @ A4 )
       => ( ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) )
          = ( minus_minus_nat @ ( finite_card_literal @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_literal @ X @ A4 )
       => ( ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) )
          = ( finite_card_literal @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4378_card__Diff__singleton__if,axiom,
    ! [X: list_nat,A4: set_list_nat] :
      ( ( ( member_list_nat @ X @ A4 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_list_nat @ X @ A4 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( finite_card_list_nat @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4379_card__Diff__singleton__if,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ( ( member_set_nat @ X @ A4 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat @ X @ A4 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
          = ( finite_card_set_nat @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4380_card__Diff__singleton__if,axiom,
    ! [X: int,A4: set_int] :
      ( ( ( member_int @ X @ A4 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X @ A4 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( finite_card_int @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4381_card__Diff__singleton__if,axiom,
    ! [X: $o,A4: set_o] :
      ( ( ( member_o @ X @ A4 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_o @ X @ A4 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( finite_card_o @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4382_card__Diff__singleton__if,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( ( member_nat @ X @ A4 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X @ A4 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( finite_card_nat @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_4383_card__Diff__singleton,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( finite7802652506058667612T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) )
        = ( minus_minus_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4384_card__Diff__singleton,axiom,
    ! [X: real,A4: set_real] :
      ( ( member_real @ X @ A4 )
     => ( ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) )
        = ( minus_minus_nat @ ( finite_card_real @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4385_card__Diff__singleton,axiom,
    ! [X: complex,A4: set_complex] :
      ( ( member_complex @ X @ A4 )
     => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
        = ( minus_minus_nat @ ( finite_card_complex @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4386_card__Diff__singleton,axiom,
    ! [X: literal,A4: set_literal] :
      ( ( member_literal @ X @ A4 )
     => ( ( finite_card_literal @ ( minus_7832829386415567259iteral @ A4 @ ( insert_literal @ X @ bot_bot_set_literal ) ) )
        = ( minus_minus_nat @ ( finite_card_literal @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4387_card__Diff__singleton,axiom,
    ! [X: list_nat,A4: set_list_nat] :
      ( ( member_list_nat @ X @ A4 )
     => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_list_nat @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4388_card__Diff__singleton,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ X @ A4 )
     => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_set_nat @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4389_card__Diff__singleton,axiom,
    ! [X: int,A4: set_int] :
      ( ( member_int @ X @ A4 )
     => ( ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) )
        = ( minus_minus_nat @ ( finite_card_int @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4390_card__Diff__singleton,axiom,
    ! [X: $o,A4: set_o] :
      ( ( member_o @ X @ A4 )
     => ( ( finite_card_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) )
        = ( minus_minus_nat @ ( finite_card_o @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4391_card__Diff__singleton,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_4392_the__default_Osimps_I1_J,axiom,
    ! [Uu2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( the_de3812365490807259288at_nat @ Uu2 @ ( some_P7363390416028606310at_nat @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_4393_the__default_Osimps_I1_J,axiom,
    ! [Uu2: nat,X: nat] :
      ( ( the_default_nat @ Uu2 @ ( some_nat @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_4394_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P7: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P7 @ ( power_power_rat @ P7 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4395_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P7: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P7 @ ( power_power_complex @ P7 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4396_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P7: assn,M5: nat] : ( if_assn @ ( M5 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P7 @ ( power_power_assn @ P7 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4397_power__eq__if,axiom,
    ( power_power_uint32
    = ( ^ [P7: uint32,M5: nat] : ( if_uint32 @ ( M5 = zero_zero_nat ) @ one_one_uint32 @ ( times_times_uint32 @ P7 @ ( power_power_uint32 @ P7 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4398_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P7: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P7 @ ( power_power_real @ P7 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4399_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P7: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P7 @ ( power_power_nat @ P7 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4400_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P7: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P7 @ ( power_power_int @ P7 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4401_the__default_Osimps_I2_J,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_de3812365490807259288at_nat @ X @ none_P5556105721700978146at_nat )
      = X ) ).

% the_default.simps(2)
thf(fact_4402_the__default_Osimps_I2_J,axiom,
    ! [X: nat] :
      ( ( the_default_nat @ X @ none_nat )
      = X ) ).

% the_default.simps(2)
thf(fact_4403_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4404_power__minus__mult,axiom,
    ! [N: nat,A: assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_assn @ ( power_power_assn @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_assn @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4405_power__minus__mult,axiom,
    ! [N: nat,A: uint32] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_uint32 @ ( power_power_uint32 @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_uint32 @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4406_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4407_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4408_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4409_finite__enumerate__step,axiom,
    ! [S3: set_nat,N: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S3 ) )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ N ) @ ( infini8530281810654367211te_nat @ S3 @ ( suc @ N ) ) ) ) ) ).

% finite_enumerate_step
thf(fact_4410_finite__enum__subset,axiom,
    ! [X6: set_nat,Y7: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
         => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
            = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
     => ( ( finite_finite_nat @ X6 )
       => ( ( finite_finite_nat @ Y7 )
         => ( ( ord_less_eq_nat @ ( finite_card_nat @ X6 ) @ ( finite_card_nat @ Y7 ) )
           => ( ord_less_eq_set_nat @ X6 @ Y7 ) ) ) ) ) ).

% finite_enum_subset
thf(fact_4411_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4412_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4413_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4414_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4415_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4416_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P2: vEBT_VEBTi > $o,N: nat] :
      ( ! [X3: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( P2 @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4417_inthall,axiom,
    ! [Xs: list_int,P2: int > $o,N: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4418_inthall,axiom,
    ! [Xs: list_set_nat,P2: set_nat > $o,N: nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( P2 @ ( nth_set_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4419_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,N: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P2 @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4420_inthall,axiom,
    ! [Xs: list_real,P2: real > $o,N: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( P2 @ ( nth_real @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4421_inthall,axiom,
    ! [Xs: list_o,P2: $o > $o,N: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( P2 @ ( nth_o @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4422_inthall,axiom,
    ! [Xs: list_nat,P2: nat > $o,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4423_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( size_s6755466524823107622T_VEBT @ ( slice_VEBT_VEBT @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_4424_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_real] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_real @ Xs ) )
       => ( ( size_size_list_real @ ( slice_real @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_4425_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_o @ Xs ) )
       => ( ( size_size_list_o @ ( slice_o @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_4426_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs ) )
       => ( ( size_size_list_nat @ ( slice_nat @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_4427_card__Min__le__sum,axiom,
    ! [A4: set_literal,F: literal > nat] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_literal @ A4 ) @ ( lattic8721135487736765967in_nat @ ( image_literal_nat @ F @ A4 ) ) ) @ ( groups8652099787943017962al_nat @ F @ A4 ) ) ) ).

% card_Min_le_sum
thf(fact_4428_card__Min__le__sum,axiom,
    ! [A4: set_list_nat,F: list_nat > nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_list_nat @ A4 ) @ ( lattic8721135487736765967in_nat @ ( image_list_nat_nat @ F @ A4 ) ) ) @ ( groups4396056296759096172at_nat @ F @ A4 ) ) ) ).

% card_Min_le_sum
thf(fact_4429_card__Min__le__sum,axiom,
    ! [A4: set_set_nat,F: set_nat > nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_set_nat @ A4 ) @ ( lattic8721135487736765967in_nat @ ( image_set_nat_nat @ F @ A4 ) ) ) @ ( groups8294997508430121362at_nat @ F @ A4 ) ) ) ).

% card_Min_le_sum
thf(fact_4430_card__Min__le__sum,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_int @ A4 ) @ ( lattic8721135487736765967in_nat @ ( image_int_nat @ F @ A4 ) ) ) @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).

% card_Min_le_sum
thf(fact_4431_card__Min__le__sum,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_complex @ A4 ) @ ( lattic8721135487736765967in_nat @ ( image_complex_nat @ F @ A4 ) ) ) @ ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ).

% card_Min_le_sum
thf(fact_4432_card__Min__le__sum,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( lattic8721135487736765967in_nat @ ( image_951025933927791156er_nat @ F @ A4 ) ) ) @ ( groups7237345082560585321er_nat @ F @ A4 ) ) ) ).

% card_Min_le_sum
thf(fact_4433_card__Min__le__sum,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_nat @ A4 ) @ ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ A4 ) ) ) @ ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ).

% card_Min_le_sum
thf(fact_4434_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ( ( power_power_real @ R4 @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_4435_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N )
              = A )
            & ! [Y5: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y5 )
                  & ( ( power_power_real @ Y5 @ N )
                    = A ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_4436_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,N: nat] :
      ( ! [X3: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X3 @ ( set_list_VEBT_VEBT2 @ Xs ) )
         => ( ( size_s6755466524823107622T_VEBT @ X3 )
            = N ) )
     => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs ) )
        = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_4437_length__mul__elem,axiom,
    ! [Xs: list_list_real,N: nat] :
      ( ! [X3: list_real] :
          ( ( member_list_real @ X3 @ ( set_list_real2 @ Xs ) )
         => ( ( size_size_list_real @ X3 )
            = N ) )
     => ( ( size_size_list_real @ ( concat_real @ Xs ) )
        = ( times_times_nat @ ( size_s6660260683639930848t_real @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_4438_length__mul__elem,axiom,
    ! [Xs: list_list_o,N: nat] :
      ( ! [X3: list_o] :
          ( ( member_list_o @ X3 @ ( set_list_o2 @ Xs ) )
         => ( ( size_size_list_o @ X3 )
            = N ) )
     => ( ( size_size_list_o @ ( concat_o @ Xs ) )
        = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_4439_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
         => ( ( size_size_list_nat @ X3 )
            = N ) )
     => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
        = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_4440_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
     => ( ( rotate1_VEBT_VEBT @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_4441_rotate1__length01,axiom,
    ! [Xs: list_real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ one_one_nat )
     => ( ( rotate1_real @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_4442_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( ( rotate1_o @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_4443_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_4444_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
      = ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_4445_slice__complete,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( slice_VEBT_VEBT @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_4446_slice__complete,axiom,
    ! [Xs: list_real] :
      ( ( slice_real @ zero_zero_nat @ ( size_size_list_real @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_4447_slice__complete,axiom,
    ! [Xs: list_o] :
      ( ( slice_o @ zero_zero_nat @ ( size_size_list_o @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_4448_slice__complete,axiom,
    ! [Xs: list_nat] :
      ( ( slice_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_4449_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_4450_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_4451_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y4: real] :
          ( ( ord_less_real @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% less_eq_real_def
thf(fact_4452_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).

% real_arch_pow
thf(fact_4453_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_4454_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_less_0_iff
thf(fact_4455_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_4456_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs )
        = ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
           => ( ( nth_VEBT_VEBTi @ Xs @ I2 )
              = ( nth_VEBT_VEBTi @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4457_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I2 )
              = ( nth_int @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4458_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I2 )
              = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4459_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
           => ( ( nth_real @ Xs @ I2 )
              = ( nth_real @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4460_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I2 )
              = ( nth_o @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4461_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4462_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > vEBT_VEBTi > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: vEBT_VEBTi] : ( P2 @ I3 @ X8 ) ) )
      = ( ? [Xs2: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_VEBT_VEBTi @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4463_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > int > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: int] : ( P2 @ I3 @ X8 ) ) )
      = ( ? [Xs2: list_int] :
            ( ( ( size_size_list_int @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4464_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: vEBT_VEBT] : ( P2 @ I3 @ X8 ) ) )
      = ( ? [Xs2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4465_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > real > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: real] : ( P2 @ I3 @ X8 ) ) )
      = ( ? [Xs2: list_real] :
            ( ( ( size_size_list_real @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_real @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4466_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: $o] : ( P2 @ I3 @ X8 ) ) )
      = ( ? [Xs2: list_o] :
            ( ( ( size_size_list_o @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4467_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: nat] : ( P2 @ I3 @ X8 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4468_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBTi,Z4: list_VEBT_VEBTi] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_VEBT_VEBTi,Ys3: list_VEBT_VEBTi] :
          ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( size_s7982070591426661849_VEBTi @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
             => ( ( nth_VEBT_VEBTi @ Xs2 @ I3 )
                = ( nth_VEBT_VEBTi @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4469_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z4: list_int] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
             => ( ( nth_int @ Xs2 @ I3 )
                = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4470_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
                = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4471_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_real,Z4: list_real] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_real,Ys3: list_real] :
          ( ( ( size_size_list_real @ Xs2 )
            = ( size_size_list_real @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
             => ( ( nth_real @ Xs2 @ I3 )
                = ( nth_real @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4472_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z4: list_o] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
             => ( ( nth_o @ Xs2 @ I3 )
                = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4473_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4474_obtain__list__from__elements,axiom,
    ! [N: nat,P2: vEBT_VEBTi > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: vEBT_VEBTi] : ( P2 @ Li @ I2 ) )
     => ~ ! [L2: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P2 @ ( nth_VEBT_VEBTi @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4475_obtain__list__from__elements,axiom,
    ! [N: nat,P2: int > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: int] : ( P2 @ Li @ I2 ) )
     => ~ ! [L2: list_int] :
            ( ( ( size_size_list_int @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P2 @ ( nth_int @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4476_obtain__list__from__elements,axiom,
    ! [N: nat,P2: vEBT_VEBT > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: vEBT_VEBT] : ( P2 @ Li @ I2 ) )
     => ~ ! [L2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P2 @ ( nth_VEBT_VEBT @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4477_obtain__list__from__elements,axiom,
    ! [N: nat,P2: real > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: real] : ( P2 @ Li @ I2 ) )
     => ~ ! [L2: list_real] :
            ( ( ( size_size_list_real @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P2 @ ( nth_real @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4478_obtain__list__from__elements,axiom,
    ! [N: nat,P2: $o > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: $o] : ( P2 @ Li @ I2 ) )
     => ~ ! [L2: list_o] :
            ( ( ( size_size_list_o @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P2 @ ( nth_o @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4479_obtain__list__from__elements,axiom,
    ! [N: nat,P2: nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: nat] : ( P2 @ Li @ I2 ) )
     => ~ ! [L2: list_nat] :
            ( ( ( size_size_list_nat @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P2 @ ( nth_nat @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4480_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBTi,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_VEBT_VEBTi @ ( slice_VEBT_VEBTi @ From @ To @ Xs ) @ I )
            = ( nth_VEBT_VEBTi @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_4481_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_int @ ( slice_int @ From @ To @ Xs ) @ I )
            = ( nth_int @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_4482_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBT,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_VEBT_VEBT @ ( slice_VEBT_VEBT @ From @ To @ Xs ) @ I )
            = ( nth_VEBT_VEBT @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_4483_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_real,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_real @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_real @ ( slice_real @ From @ To @ Xs ) @ I )
            = ( nth_real @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_4484_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_o,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_o @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_o @ ( slice_o @ From @ To @ Xs ) @ I )
            = ( nth_o @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_4485_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_nat @ ( slice_nat @ From @ To @ Xs ) @ I )
            = ( nth_nat @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_4486_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P2: vEBT_VEBTi > $o] :
      ( ( ! [X4: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ L ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) )
           => ( P2 @ ( nth_VEBT_VEBTi @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4487_all__set__conv__nth,axiom,
    ! [L: list_int,P2: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ L ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) )
           => ( P2 @ ( nth_int @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4488_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ L ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) )
           => ( P2 @ ( nth_VEBT_VEBT @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4489_all__set__conv__nth,axiom,
    ! [L: list_real,P2: real > $o] :
      ( ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ L ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ L ) )
           => ( P2 @ ( nth_real @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4490_all__set__conv__nth,axiom,
    ! [L: list_o,P2: $o > $o] :
      ( ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ L ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ L ) )
           => ( P2 @ ( nth_o @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4491_all__set__conv__nth,axiom,
    ! [L: list_nat,P2: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ L ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
           => ( P2 @ ( nth_nat @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4492_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( member_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ N ) @ ( set_VEBT_VEBTi2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4493_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4494_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4495_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4496_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4497_nth__mem,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4498_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4499_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,P2: vEBT_VEBTi > $o] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ! [X3: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4500_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P2: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4501_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4502_list__ball__nth,axiom,
    ! [N: nat,Xs: list_real,P2: real > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_real @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4503_list__ball__nth,axiom,
    ! [N: nat,Xs: list_o,P2: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4504_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P2: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4505_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
            & ( ( nth_VEBT_VEBTi @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4506_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4507_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4508_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4509_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4510_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4511_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4512_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P2: vEBT_VEBTi > $o,X: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
         => ( P2 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4513_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P2: int > $o,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
         => ( P2 @ ( nth_int @ Xs @ I2 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4514_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P2: set_nat > $o,X: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
         => ( P2 @ ( nth_set_nat @ Xs @ I2 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4515_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4516_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P2: real > $o,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
         => ( P2 @ ( nth_real @ Xs @ I2 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4517_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P2: $o > $o,X: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
         => ( P2 @ ( nth_o @ Xs @ I2 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4518_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P2 @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4519_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P2: vEBT_VEBTi > $o] :
      ( ( ! [X4: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
           => ( P2 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4520_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( P2 @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4521_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4522_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P2: real > $o] :
      ( ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
           => ( P2 @ ( nth_real @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4523_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P2: $o > $o] :
      ( ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
           => ( P2 @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4524_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( P2 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4525_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R4: real] :
          ( ( ord_less_real @ zero_zero_real @ R4 )
          & ( ( power_power_real @ R4 @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_4526_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_4527_set__image__eq__pointwiseI,axiom,
    ! [L: list_int,L3: list_int,F: int > nat] :
      ( ( ( size_size_list_int @ L )
        = ( size_size_list_int @ L3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ L ) )
           => ( ( F @ ( nth_int @ L @ I2 ) )
              = ( F @ ( nth_int @ L3 @ I2 ) ) ) )
       => ( ( image_int_nat @ F @ ( set_int2 @ L ) )
          = ( image_int_nat @ F @ ( set_int2 @ L3 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_4528_set__image__eq__pointwiseI,axiom,
    ! [L: list_int,L3: list_int,F: int > int] :
      ( ( ( size_size_list_int @ L )
        = ( size_size_list_int @ L3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ L ) )
           => ( ( F @ ( nth_int @ L @ I2 ) )
              = ( F @ ( nth_int @ L3 @ I2 ) ) ) )
       => ( ( image_int_int @ F @ ( set_int2 @ L ) )
          = ( image_int_int @ F @ ( set_int2 @ L3 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_4529_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L3: list_nat,F: nat > set_nat] :
      ( ( ( size_size_list_nat @ L )
        = ( size_size_list_nat @ L3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( ( F @ ( nth_nat @ L @ I2 ) )
              = ( F @ ( nth_nat @ L3 @ I2 ) ) ) )
       => ( ( image_nat_set_nat @ F @ ( set_nat2 @ L ) )
          = ( image_nat_set_nat @ F @ ( set_nat2 @ L3 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_4530_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L3: list_nat,F: nat > nat] :
      ( ( ( size_size_list_nat @ L )
        = ( size_size_list_nat @ L3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( ( F @ ( nth_nat @ L @ I2 ) )
              = ( F @ ( nth_nat @ L3 @ I2 ) ) ) )
       => ( ( image_nat_nat @ F @ ( set_nat2 @ L ) )
          = ( image_nat_nat @ F @ ( set_nat2 @ L3 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_4531_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L3: list_nat,F: nat > int] :
      ( ( ( size_size_list_nat @ L )
        = ( size_size_list_nat @ L3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( ( F @ ( nth_nat @ L @ I2 ) )
              = ( F @ ( nth_nat @ L3 @ I2 ) ) ) )
       => ( ( image_nat_int @ F @ ( set_nat2 @ L ) )
          = ( image_nat_int @ F @ ( set_nat2 @ L3 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_4532_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > nat,X: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_nat @ ( F @ X ) @ ( image_VEBT_VEBTi_nat @ F @ ( set_VEBT_VEBTi2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) )
            & ( ( F @ ( nth_VEBT_VEBTi @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4533_in__set__image__conv__nth,axiom,
    ! [F: int > nat,X: int,L: list_int] :
      ( ( member_nat @ ( F @ X ) @ ( image_int_nat @ F @ ( set_int2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) )
            & ( ( F @ ( nth_int @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4534_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > vEBT_VEBT,X: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBT @ ( F @ X ) @ ( image_7547481670047419768T_VEBT @ F @ ( set_VEBT_VEBTi2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) )
            & ( ( F @ ( nth_VEBT_VEBTi @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4535_in__set__image__conv__nth,axiom,
    ! [F: int > vEBT_VEBT,X: int,L: list_int] :
      ( ( member_VEBT_VEBT @ ( F @ X ) @ ( image_int_VEBT_VEBT @ F @ ( set_int2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) )
            & ( ( F @ ( nth_int @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4536_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > real,X: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_real @ ( F @ X ) @ ( image_6202559892754154600i_real @ F @ ( set_VEBT_VEBTi2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) )
            & ( ( F @ ( nth_VEBT_VEBTi @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4537_in__set__image__conv__nth,axiom,
    ! [F: int > real,X: int,L: list_int] :
      ( ( member_real @ ( F @ X ) @ ( image_int_real @ F @ ( set_int2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) )
            & ( ( F @ ( nth_int @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4538_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > int,X: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_int @ ( F @ X ) @ ( image_VEBT_VEBTi_int @ F @ ( set_VEBT_VEBTi2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) )
            & ( ( F @ ( nth_VEBT_VEBTi @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4539_in__set__image__conv__nth,axiom,
    ! [F: int > int,X: int,L: list_int] :
      ( ( member_int @ ( F @ X ) @ ( image_int_int @ F @ ( set_int2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) )
            & ( ( F @ ( nth_int @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4540_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBT > nat,X: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( ( member_nat @ ( F @ X ) @ ( image_VEBT_VEBT_nat @ F @ ( set_VEBT_VEBT2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) )
            & ( ( F @ ( nth_VEBT_VEBT @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4541_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,X: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ ( F @ X ) @ ( image_3375948659692109573T_VEBT @ F @ ( set_VEBT_VEBT2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) )
            & ( ( F @ ( nth_VEBT_VEBT @ L @ I3 ) )
              = ( F @ X ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_4542_sum__count__set,axiom,
    ! [Xs: list_complex,X6: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X6 )
     => ( ( finite3207457112153483333omplex @ X6 )
       => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X6 )
          = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_4543_sum__count__set,axiom,
    ! [Xs: list_Code_integer,X6: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ X6 )
     => ( ( finite6017078050557962740nteger @ X6 )
       => ( ( groups7237345082560585321er_nat @ ( count_3970941599679287265nteger @ Xs ) @ X6 )
          = ( size_s3445333598471063425nteger @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_4544_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X6 )
     => ( ( finite5795047828879050333T_VEBT @ X6 )
       => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X6 )
          = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_4545_sum__count__set,axiom,
    ! [Xs: list_real,X6: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ X6 )
     => ( ( finite_finite_real @ X6 )
       => ( ( groups1935376822645274424al_nat @ ( count_list_real @ Xs ) @ X6 )
          = ( size_size_list_real @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_4546_sum__count__set,axiom,
    ! [Xs: list_o,X6: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X6 )
     => ( ( finite_finite_o @ X6 )
       => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X6 )
          = ( size_size_list_o @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_4547_sum__count__set,axiom,
    ! [Xs: list_int,X6: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X6 )
     => ( ( finite_finite_int @ X6 )
       => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X6 )
          = ( size_size_list_int @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_4548_sum__count__set,axiom,
    ! [Xs: list_nat,X6: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X6 )
     => ( ( finite_finite_nat @ X6 )
       => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X6 )
          = ( size_size_list_nat @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_4549_sum__le__card__Max,axiom,
    ! [A4: set_literal,F: literal > nat] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ord_less_eq_nat @ ( groups8652099787943017962al_nat @ F @ A4 ) @ ( times_times_nat @ ( finite_card_literal @ A4 ) @ ( lattic8265883725875713057ax_nat @ ( image_literal_nat @ F @ A4 ) ) ) ) ) ).

% sum_le_card_Max
thf(fact_4550_sum__le__card__Max,axiom,
    ! [A4: set_list_nat,F: list_nat > nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ A4 ) @ ( times_times_nat @ ( finite_card_list_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ ( image_list_nat_nat @ F @ A4 ) ) ) ) ) ).

% sum_le_card_Max
thf(fact_4551_sum__le__card__Max,axiom,
    ! [A4: set_set_nat,F: set_nat > nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( times_times_nat @ ( finite_card_set_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ ( image_set_nat_nat @ F @ A4 ) ) ) ) ) ).

% sum_le_card_Max
thf(fact_4552_sum__le__card__Max,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( times_times_nat @ ( finite_card_int @ A4 ) @ ( lattic8265883725875713057ax_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ).

% sum_le_card_Max
thf(fact_4553_sum__le__card__Max,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( times_times_nat @ ( finite_card_complex @ A4 ) @ ( lattic8265883725875713057ax_nat @ ( image_complex_nat @ F @ A4 ) ) ) ) ) ).

% sum_le_card_Max
thf(fact_4554_sum__le__card__Max,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ord_less_eq_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( times_times_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( lattic8265883725875713057ax_nat @ ( image_951025933927791156er_nat @ F @ A4 ) ) ) ) ) ).

% sum_le_card_Max
thf(fact_4555_sum__le__card__Max,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( times_times_nat @ ( finite_card_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ).

% sum_le_card_Max
thf(fact_4556_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X @ A4 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8691415230153176458o_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4557_sum_Oinsert,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ~ ( member_VEBT_VEBT @ X @ A4 )
       => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4558_sum_Oinsert,axiom,
    ! [A4: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A4 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4559_sum_Oinsert,axiom,
    ! [A4: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A4 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4560_sum_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4561_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ~ ( member_Code_integer @ X @ A4 )
       => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A4 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4562_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4563_sum_Oinsert,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ~ ( member_VEBT_VEBT @ X @ A4 )
       => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4564_sum_Oinsert,axiom,
    ! [A4: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4565_sum_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_4566_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > uint32] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups833757482993574392uint32 @ G @ A4 )
        = zero_zero_uint32 ) ) ).

% sum.infinite
thf(fact_4567_sum_Oinfinite,axiom,
    ! [A4: set_int,G: int > uint32] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups5712668689793887828uint32 @ G @ A4 )
        = zero_zero_uint32 ) ) ).

% sum.infinite
thf(fact_4568_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > uint32] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups8736914816313324502uint32 @ G @ A4 )
        = zero_zero_uint32 ) ) ).

% sum.infinite
thf(fact_4569_sum_Oinfinite,axiom,
    ! [A4: set_Code_integer,G: code_integer > uint32] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ( ( groups8847630953604152069uint32 @ G @ A4 )
        = zero_zero_uint32 ) ) ).

% sum.infinite
thf(fact_4570_sum_Oinfinite,axiom,
    ! [A4: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_4571_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_4572_sum_Oinfinite,axiom,
    ! [A4: set_Code_integer,G: code_integer > real] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1270011288395367621r_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_4573_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_4574_sum_Oinfinite,axiom,
    ! [A4: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_4575_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_4576_sum__eq__0__iff,axiom,
    ! [F2: set_int,F: int > nat] :
      ( ( finite_finite_int @ F2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ F2 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4577_sum__eq__0__iff,axiom,
    ! [F2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ F2 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4578_sum__eq__0__iff,axiom,
    ! [F2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ F2 )
     => ( ( ( groups7237345082560585321er_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ F2 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4579_sum__eq__0__iff,axiom,
    ! [F2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ F2 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4580_sum_Oempty,axiom,
    ! [G: nat > uint32] :
      ( ( groups833757482993574392uint32 @ G @ bot_bot_set_nat )
      = zero_zero_uint32 ) ).

% sum.empty
thf(fact_4581_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_4582_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_4583_sum_Oempty,axiom,
    ! [G: int > uint32] :
      ( ( groups5712668689793887828uint32 @ G @ bot_bot_set_int )
      = zero_zero_uint32 ) ).

% sum.empty
thf(fact_4584_sum_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
      = zero_zero_real ) ).

% sum.empty
thf(fact_4585_sum_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_4586_sum_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_4587_sum_Oempty,axiom,
    ! [G: $o > uint32] :
      ( ( groups7241207224191747786uint32 @ G @ bot_bot_set_o )
      = zero_zero_uint32 ) ).

% sum.empty
thf(fact_4588_sum_Oempty,axiom,
    ! [G: $o > real] :
      ( ( groups8691415230153176458o_real @ G @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty
thf(fact_4589_sum_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups7872700643590313910_o_rat @ G @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_4590_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( member_VEBT_VEBT @ I @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups2240296850493347238T_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4591_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: real > real] :
      ( ( member_real @ I @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4592_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4593_member__le__sum,axiom,
    ! [I: code_integer,A4: set_Code_integer,F: code_integer > real] :
      ( ( member_Code_integer @ I @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups1270011288395367621r_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4594_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: int > real] :
      ( ( member_int @ I @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4595_member__le__sum,axiom,
    ! [I: $o,A4: set_o,F: $o > real] :
      ( ( member_o @ I @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( minus_minus_set_o @ A4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_o @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8691415230153176458o_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4596_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( member_VEBT_VEBT @ I @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups136491112297645522BT_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4597_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: real > rat] :
      ( ( member_real @ I @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4598_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4599_member__le__sum,axiom,
    ! [I: code_integer,A4: set_Code_integer,F: code_integer > rat] :
      ( ( member_Code_integer @ I @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups6602215022474089585er_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_4600_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( groups2240296850493347238T_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4601_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4602_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4603_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,B: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4604_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ ( groups136491112297645522BT_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4605_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4606_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4607_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,B: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4608_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4609_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B4 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B4 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) @ ( groups771621172384141258BT_nat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4610_sum__diff1,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( ( member_VEBT_VEBT @ A @ A4 )
         => ( ( groups8325533452322294502uint32 @ F @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_uint32 @ ( groups8325533452322294502uint32 @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A4 )
         => ( ( groups8325533452322294502uint32 @ F @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups8325533452322294502uint32 @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4611_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: real > uint32] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ A @ A4 )
         => ( ( groups5944083974425963860uint32 @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_uint32 @ ( groups5944083974425963860uint32 @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A4 )
         => ( ( groups5944083974425963860uint32 @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups5944083974425963860uint32 @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4612_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: complex > uint32] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ A @ A4 )
         => ( ( groups8736914816313324502uint32 @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_uint32 @ ( groups8736914816313324502uint32 @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A4 )
         => ( ( groups8736914816313324502uint32 @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups8736914816313324502uint32 @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4613_sum__diff1,axiom,
    ! [A4: set_Code_integer,A: code_integer,F: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( member_Code_integer @ A @ A4 )
         => ( ( groups8847630953604152069uint32 @ F @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_uint32 @ ( groups8847630953604152069uint32 @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Code_integer @ A @ A4 )
         => ( ( groups8847630953604152069uint32 @ F @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( groups8847630953604152069uint32 @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4614_sum__diff1,axiom,
    ! [A4: set_int,A: int,F: int > uint32] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ A @ A4 )
         => ( ( groups5712668689793887828uint32 @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_uint32 @ ( groups5712668689793887828uint32 @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A4 )
         => ( ( groups5712668689793887828uint32 @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups5712668689793887828uint32 @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4615_sum__diff1,axiom,
    ! [A4: set_o,A: $o,F: $o > uint32] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ A @ A4 )
         => ( ( groups7241207224191747786uint32 @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( minus_minus_uint32 @ ( groups7241207224191747786uint32 @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_o @ A @ A4 )
         => ( ( groups7241207224191747786uint32 @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( groups7241207224191747786uint32 @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4616_sum__diff1,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( ( member_VEBT_VEBT @ A @ A4 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A4 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups2240296850493347238T_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4617_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ A @ A4 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A4 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4618_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ A @ A4 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A4 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4619_sum__diff1,axiom,
    ! [A4: set_Code_integer,A: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( member_Code_integer @ A @ A4 )
         => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Code_integer @ A @ A4 )
         => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( groups1270011288395367621r_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_4620_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4621_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4622_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: code_integer > real,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A4 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4623_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > rat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4624_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A4 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4625_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A4 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4626_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > nat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4627_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A4 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4628_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: code_integer > nat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A4 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4629_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > int,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups769130701875090982BT_int @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_4630_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X5: real] : ( member_real @ X5 @ S3 )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S3 )
           => ( ord_less_eq_real @ X3 @ Z5 ) )
       => ? [Y3: real] :
            ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ord_less_eq_real @ X5 @ Y3 ) )
            & ! [Z5: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S3 )
                   => ( ord_less_eq_real @ X3 @ Z5 ) )
               => ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_4631_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_4632_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A4 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A4 )
          = ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_4633_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A4 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A4 )
          = ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_4634_sum_Oneutral,axiom,
    ! [A4: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A4 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_4635_sum_Oneutral,axiom,
    ! [A4: set_complex,G: complex > complex] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A4 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_4636_sum_Oneutral,axiom,
    ! [A4: set_nat,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_4637_sum_Oneutral,axiom,
    ! [A4: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A4 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_4638_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > uint32,A4: set_nat] :
      ( ( ( groups833757482993574392uint32 @ G @ A4 )
       != zero_zero_uint32 )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4639_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > uint32,A4: set_VEBT_VEBT] :
      ( ( ( groups8325533452322294502uint32 @ G @ A4 )
       != zero_zero_uint32 )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4640_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > uint32,A4: set_real] :
      ( ( ( groups5944083974425963860uint32 @ G @ A4 )
       != zero_zero_uint32 )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4641_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > uint32,A4: set_int] :
      ( ( ( groups5712668689793887828uint32 @ G @ A4 )
       != zero_zero_uint32 )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4642_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > real,A4: set_VEBT_VEBT] :
      ( ( ( groups2240296850493347238T_real @ G @ A4 )
       != zero_zero_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4643_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A4: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A4 )
       != zero_zero_real )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4644_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A4: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A4 )
       != zero_zero_real )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4645_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A4: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A4 )
       != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4646_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > rat,A4: set_VEBT_VEBT] :
      ( ( ( groups136491112297645522BT_rat @ G @ A4 )
       != zero_zero_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4647_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A4: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A4 )
       != zero_zero_rat )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G @ A3 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_4648_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_4649_sum__nonpos,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_4650_sum__nonpos,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_4651_sum__nonpos,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4652_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4653_sum__nonpos,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4654_sum__nonpos,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4655_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_4656_sum__nonpos,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_4657_sum__nonpos,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_4658_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4659_sum__nonneg,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4660_sum__nonneg,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4661_sum__nonneg,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4662_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4663_sum__nonneg,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4664_sum__nonneg,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4665_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4666_sum__nonneg,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4667_sum__nonneg,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4668_sum__mono__inv,axiom,
    ! [F: vEBT_VEBT > rat,I5: set_VEBT_VEBT,G: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( ( groups136491112297645522BT_rat @ F @ I5 )
        = ( groups136491112297645522BT_rat @ G @ I5 ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4669_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4670_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4671_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4672_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4673_sum__mono__inv,axiom,
    ! [F: code_integer > rat,I5: set_Code_integer,G: code_integer > rat,I: code_integer] :
      ( ( ( groups6602215022474089585er_rat @ F @ I5 )
        = ( groups6602215022474089585er_rat @ G @ I5 ) )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_Code_integer @ I @ I5 )
         => ( ( finite6017078050557962740nteger @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4674_sum__mono__inv,axiom,
    ! [F: vEBT_VEBT > nat,I5: set_VEBT_VEBT,G: vEBT_VEBT > nat,I: vEBT_VEBT] :
      ( ( ( groups771621172384141258BT_nat @ F @ I5 )
        = ( groups771621172384141258BT_nat @ G @ I5 ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4675_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4676_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4677_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4678_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4679_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4680_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4681_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4682_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4683_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4684_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4685_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4686_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4687_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4688_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4689_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4690_sum__le__included,axiom,
    ! [S: set_int,T: set_Code_integer,G: code_integer > real,I: code_integer > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: code_integer] :
                    ( ( member_Code_integer @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4691_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4692_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4693_sum__le__included,axiom,
    ! [S: set_complex,T: set_Code_integer,G: code_integer > real,I: code_integer > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: code_integer] :
                    ( ( member_Code_integer @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4694_sum__le__included,axiom,
    ! [S: set_Code_integer,T: set_int,G: int > real,I: int > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( 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 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4695_sum__le__included,axiom,
    ! [S: set_Code_integer,T: set_complex,G: complex > real,I: complex > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4696_sum__le__included,axiom,
    ! [S: set_Code_integer,T: set_Code_integer,G: code_integer > real,I: code_integer > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( 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 @ S )
               => ? [Xa: code_integer] :
                    ( ( member_Code_integer @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4697_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4698_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A4 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4699_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4700_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ A4 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4701_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4702_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A4 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4703_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4704_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ A4 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4705_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A4 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4706_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4707_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ A4 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4708_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S3: set_nat,H2: nat > uint32,G: nat > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_uint32 @ X15 @ Y15 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups833757482993574392uint32 @ H2 @ S3 ) @ ( groups833757482993574392uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4709_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S3: set_int,H2: int > uint32,G: int > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_uint32 @ X15 @ Y15 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5712668689793887828uint32 @ H2 @ S3 ) @ ( groups5712668689793887828uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4710_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S3: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_uint32 @ X15 @ Y15 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups8736914816313324502uint32 @ H2 @ S3 ) @ ( groups8736914816313324502uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4711_sum_Orelated,axiom,
    ! [R2: uint32 > uint32 > $o,S3: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( R2 @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_uint32 @ X15 @ Y15 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S3 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups8847630953604152069uint32 @ H2 @ S3 ) @ ( groups8847630953604152069uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4712_sum_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4713_sum_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4714_sum_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S3 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups1270011288395367621r_real @ H2 @ S3 ) @ ( groups1270011288395367621r_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4715_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4716_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4717_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4718_sum__strict__mono,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( A4 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( groups2240296850493347238T_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4719_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4720_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4721_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4722_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4723_sum__strict__mono,axiom,
    ! [A4: set_o,F: $o > real,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A4 ) @ ( groups8691415230153176458o_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4724_sum__strict__mono,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( A4 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ ( groups136491112297645522BT_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4725_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4726_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4727_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4728_sum_Oinsert__if,axiom,
    ! [A4: set_o,X: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X @ A4 )
         => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X @ A4 ) )
            = ( groups8691415230153176458o_real @ G @ A4 ) ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X @ A4 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8691415230153176458o_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4729_sum_Oinsert__if,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( ( member_VEBT_VEBT @ X @ A4 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
            = ( groups2240296850493347238T_real @ G @ A4 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A4 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4730_sum_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X @ A4 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A4 ) )
            = ( groups8097168146408367636l_real @ G @ A4 ) ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A4 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4731_sum_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A4 ) )
            = ( groups8778361861064173332t_real @ G @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A4 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4732_sum_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X @ A4 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
            = ( groups5808333547571424918x_real @ G @ A4 ) ) )
        & ( ~ ( member_complex @ X @ A4 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4733_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( member_Code_integer @ X @ A4 )
         => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( groups1270011288395367621r_real @ G @ A4 ) ) )
        & ( ~ ( member_Code_integer @ X @ A4 )
         => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4734_sum_Oinsert__if,axiom,
    ! [A4: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X @ A4 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A4 ) )
            = ( groups7872700643590313910_o_rat @ G @ A4 ) ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4735_sum_Oinsert__if,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( ( member_VEBT_VEBT @ X @ A4 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
            = ( groups136491112297645522BT_rat @ G @ A4 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A4 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4736_sum_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X @ A4 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A4 ) )
            = ( groups1300246762558778688al_rat @ G @ A4 ) ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4737_sum_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X @ A4 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A4 ) )
            = ( groups2906978787729119204at_rat @ G @ A4 ) ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_4738_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T5: set_VEBT_VEBT,S3: set_VEBT_VEBT,I: vEBT_VEBT > vEBT_VEBT,J: vEBT_VEBT > vEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T3 @ T5 ) ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ T3 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B3 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups8325533452322294502uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4739_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T5: set_real,S3: set_VEBT_VEBT,I: real > vEBT_VEBT,J: vEBT_VEBT > real,T3: set_real,G: vEBT_VEBT > uint32,H2: real > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B3 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups5944083974425963860uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4740_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T5: set_VEBT_VEBT,S3: set_real,I: vEBT_VEBT > real,J: real > vEBT_VEBT,T3: set_VEBT_VEBT,G: real > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T3 @ T5 ) ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups8325533452322294502uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4741_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T5: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups5944083974425963860uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4742_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T5: set_int,S3: set_VEBT_VEBT,I: int > vEBT_VEBT,J: vEBT_VEBT > int,T3: set_int,G: vEBT_VEBT > uint32,H2: int > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B3 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups5712668689793887828uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4743_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T5: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > uint32,H2: int > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups5712668689793887828uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4744_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T5: set_complex,S3: set_VEBT_VEBT,I: complex > vEBT_VEBT,J: vEBT_VEBT > complex,T3: set_complex,G: vEBT_VEBT > uint32,H2: complex > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B3 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups8736914816313324502uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4745_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T5: set_complex,S3: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > uint32,H2: complex > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups8736914816313324502uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4746_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T5: set_Code_integer,S3: set_VEBT_VEBT,I: code_integer > vEBT_VEBT,J: vEBT_VEBT > code_integer,T3: set_Code_integer,G: vEBT_VEBT > uint32,H2: code_integer > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T3 @ T5 ) ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: code_integer] :
                    ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ T3 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B3 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: code_integer] :
                        ( ( member_Code_integer @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups8847630953604152069uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4747_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T5: set_Code_integer,S3: set_real,I: code_integer > real,J: real > code_integer,T3: set_Code_integer,G: real > uint32,H2: code_integer > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T3 @ T5 ) ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: code_integer] :
                    ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B3: code_integer] :
                        ( ( member_Code_integer @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups8847630953604152069uint32 @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4748_sum__SucD,axiom,
    ! [F: nat > nat,A4: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A4 )
        = ( suc @ N ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_4749_sum__eq__Suc0__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y4: int] :
                  ( ( member_int @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_4750_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y4: complex] :
                  ( ( member_complex @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_4751_sum__eq__Suc0__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( groups7237345082560585321er_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y4: code_integer] :
                  ( ( member_Code_integer @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_4752_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_4753_sum__eq__1__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
          = one_one_nat )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y4: int] :
                  ( ( member_int @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_4754_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
          = one_one_nat )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y4: complex] :
                  ( ( member_complex @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_4755_sum__eq__1__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( groups7237345082560585321er_nat @ F @ A4 )
          = one_one_nat )
        = ( ? [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A4 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y4: code_integer] :
                  ( ( member_Code_integer @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_4756_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A4 )
          = one_one_nat )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_4757_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4758_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4759_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4760_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4761_sum__pos2,axiom,
    ! [I5: set_Code_integer,I: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4762_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4763_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4764_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4765_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4766_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4767_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4768_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4769_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4770_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4771_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4772_sum__pos,axiom,
    ! [I5: set_o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4773_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4774_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4775_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4776_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups6602215022474089585er_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_4777_sum_Osame__carrier,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C3 )
       => ( ( ord_le4337996190870823476T_VEBT @ B4 @ C3 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8325533452322294502uint32 @ G @ A4 )
                  = ( groups8325533452322294502uint32 @ H2 @ B4 ) )
                = ( ( groups8325533452322294502uint32 @ G @ C3 )
                  = ( groups8325533452322294502uint32 @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4778_sum_Osame__carrier,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C3 )
     => ( ( ord_less_eq_set_real @ A4 @ C3 )
       => ( ( ord_less_eq_set_real @ B4 @ C3 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups5944083974425963860uint32 @ G @ A4 )
                  = ( groups5944083974425963860uint32 @ H2 @ B4 ) )
                = ( ( groups5944083974425963860uint32 @ G @ C3 )
                  = ( groups5944083974425963860uint32 @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4779_sum_Osame__carrier,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C3 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C3 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C3 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8736914816313324502uint32 @ G @ A4 )
                  = ( groups8736914816313324502uint32 @ H2 @ B4 ) )
                = ( ( groups8736914816313324502uint32 @ G @ C3 )
                  = ( groups8736914816313324502uint32 @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4780_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B4 @ C3 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8847630953604152069uint32 @ G @ A4 )
                  = ( groups8847630953604152069uint32 @ H2 @ B4 ) )
                = ( ( groups8847630953604152069uint32 @ G @ C3 )
                  = ( groups8847630953604152069uint32 @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4781_sum_Osame__carrier,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C3 )
       => ( ( ord_le4337996190870823476T_VEBT @ B4 @ C3 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ A4 )
                  = ( groups2240296850493347238T_real @ H2 @ B4 ) )
                = ( ( groups2240296850493347238T_real @ G @ C3 )
                  = ( groups2240296850493347238T_real @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4782_sum_Osame__carrier,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C3 )
     => ( ( ord_less_eq_set_real @ A4 @ C3 )
       => ( ( ord_less_eq_set_real @ B4 @ C3 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B4 ) )
                = ( ( groups8097168146408367636l_real @ G @ C3 )
                  = ( groups8097168146408367636l_real @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4783_sum_Osame__carrier,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C3 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C3 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C3 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B4 ) )
                = ( ( groups5808333547571424918x_real @ G @ C3 )
                  = ( groups5808333547571424918x_real @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4784_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B4 @ C3 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ A4 )
                  = ( groups1270011288395367621r_real @ H2 @ B4 ) )
                = ( ( groups1270011288395367621r_real @ G @ C3 )
                  = ( groups1270011288395367621r_real @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4785_sum_Osame__carrier,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C3 )
       => ( ( ord_le4337996190870823476T_VEBT @ B4 @ C3 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ A4 )
                  = ( groups136491112297645522BT_rat @ H2 @ B4 ) )
                = ( ( groups136491112297645522BT_rat @ G @ C3 )
                  = ( groups136491112297645522BT_rat @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4786_sum_Osame__carrier,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C3 )
     => ( ( ord_less_eq_set_real @ A4 @ C3 )
       => ( ( ord_less_eq_set_real @ B4 @ C3 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B4 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C3 )
                  = ( groups1300246762558778688al_rat @ H2 @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4787_sum_Osame__carrierI,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C3 )
       => ( ( ord_le4337996190870823476T_VEBT @ B4 @ C3 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8325533452322294502uint32 @ G @ C3 )
                  = ( groups8325533452322294502uint32 @ H2 @ C3 ) )
               => ( ( groups8325533452322294502uint32 @ G @ A4 )
                  = ( groups8325533452322294502uint32 @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4788_sum_Osame__carrierI,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C3 )
     => ( ( ord_less_eq_set_real @ A4 @ C3 )
       => ( ( ord_less_eq_set_real @ B4 @ C3 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups5944083974425963860uint32 @ G @ C3 )
                  = ( groups5944083974425963860uint32 @ H2 @ C3 ) )
               => ( ( groups5944083974425963860uint32 @ G @ A4 )
                  = ( groups5944083974425963860uint32 @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4789_sum_Osame__carrierI,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C3 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C3 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C3 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8736914816313324502uint32 @ G @ C3 )
                  = ( groups8736914816313324502uint32 @ H2 @ C3 ) )
               => ( ( groups8736914816313324502uint32 @ G @ A4 )
                  = ( groups8736914816313324502uint32 @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4790_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B4 @ C3 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8847630953604152069uint32 @ G @ C3 )
                  = ( groups8847630953604152069uint32 @ H2 @ C3 ) )
               => ( ( groups8847630953604152069uint32 @ G @ A4 )
                  = ( groups8847630953604152069uint32 @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4791_sum_Osame__carrierI,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C3 )
       => ( ( ord_le4337996190870823476T_VEBT @ B4 @ C3 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ C3 )
                  = ( groups2240296850493347238T_real @ H2 @ C3 ) )
               => ( ( groups2240296850493347238T_real @ G @ A4 )
                  = ( groups2240296850493347238T_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4792_sum_Osame__carrierI,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C3 )
     => ( ( ord_less_eq_set_real @ A4 @ C3 )
       => ( ( ord_less_eq_set_real @ B4 @ C3 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C3 )
                  = ( groups8097168146408367636l_real @ H2 @ C3 ) )
               => ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4793_sum_Osame__carrierI,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C3 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C3 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C3 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C3 )
                  = ( groups5808333547571424918x_real @ H2 @ C3 ) )
               => ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4794_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B4 @ C3 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ C3 )
                  = ( groups1270011288395367621r_real @ H2 @ C3 ) )
               => ( ( groups1270011288395367621r_real @ G @ A4 )
                  = ( groups1270011288395367621r_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4795_sum_Osame__carrierI,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C3 )
       => ( ( ord_le4337996190870823476T_VEBT @ B4 @ C3 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ C3 )
                  = ( groups136491112297645522BT_rat @ H2 @ C3 ) )
               => ( ( groups136491112297645522BT_rat @ G @ A4 )
                  = ( groups136491112297645522BT_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4796_sum_Osame__carrierI,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C3 )
     => ( ( ord_less_eq_set_real @ A4 @ C3 )
       => ( ( ord_less_eq_set_real @ B4 @ C3 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C3 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C3 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C3 )
                  = ( groups1300246762558778688al_rat @ H2 @ C3 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4797_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8736914816313324502uint32 @ G @ S3 )
            = ( groups8736914816313324502uint32 @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4798_sum_Omono__neutral__left,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8847630953604152069uint32 @ G @ S3 )
            = ( groups8847630953604152069uint32 @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4799_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S3 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4800_sum_Omono__neutral__left,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ S3 )
            = ( groups1270011288395367621r_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4801_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S3 )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4802_sum_Omono__neutral__left,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ S3 )
            = ( groups6602215022474089585er_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4803_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S3 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4804_sum_Omono__neutral__left,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ S3 )
            = ( groups7237345082560585321er_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4805_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S3 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4806_sum_Omono__neutral__left,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ S3 )
            = ( groups7234854612051535045er_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4807_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8736914816313324502uint32 @ G @ T3 )
            = ( groups8736914816313324502uint32 @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4808_sum_Omono__neutral__right,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8847630953604152069uint32 @ G @ T3 )
            = ( groups8847630953604152069uint32 @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4809_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4810_sum_Omono__neutral__right,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ T3 )
            = ( groups1270011288395367621r_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4811_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4812_sum_Omono__neutral__right,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ T3 )
            = ( groups6602215022474089585er_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4813_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4814_sum_Omono__neutral__right,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ T3 )
            = ( groups7237345082560585321er_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4815_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4816_sum_Omono__neutral__right,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ T3 )
            = ( groups7234854612051535045er_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4817_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > uint32,G: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T3 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8325533452322294502uint32 @ G @ S3 )
              = ( groups8325533452322294502uint32 @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4818_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > uint32,G: real > uint32] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5944083974425963860uint32 @ G @ S3 )
              = ( groups5944083974425963860uint32 @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4819_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8736914816313324502uint32 @ G @ S3 )
              = ( groups8736914816313324502uint32 @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4820_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8847630953604152069uint32 @ G @ S3 )
              = ( groups8847630953604152069uint32 @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4821_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T3 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ S3 )
              = ( groups2240296850493347238T_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4822_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4823_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S3 )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4824_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ S3 )
              = ( groups1270011288395367621r_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4825_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T3 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ S3 )
              = ( groups136491112297645522BT_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4826_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S3 )
              = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4827_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T3 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8325533452322294502uint32 @ G @ T3 )
              = ( groups8325533452322294502uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4828_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5944083974425963860uint32 @ G @ T3 )
              = ( groups5944083974425963860uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4829_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8736914816313324502uint32 @ G @ T3 )
              = ( groups8736914816313324502uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4830_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8847630953604152069uint32 @ G @ T3 )
              = ( groups8847630953604152069uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4831_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T3 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ T3 )
              = ( groups2240296850493347238T_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4832_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4833_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4834_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Code_integer,S3: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T3 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ T3 )
              = ( groups1270011288395367621r_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4835_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T3 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ T3 )
              = ( groups136491112297645522BT_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4836_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4837_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ A4 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5808333547571424918x_real @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4838_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: code_integer > real] :
      ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups1270011288395367621r_real @ G @ A4 )
          = ( plus_plus_real @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) @ ( groups1270011288395367621r_real @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4839_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5058264527183730370ex_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4840_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups6602215022474089585er_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) @ ( groups6602215022474089585er_rat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4841_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5693394587270226106ex_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4842_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups7237345082560585321er_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) @ ( groups7237345082560585321er_nat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4843_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5690904116761175830ex_int @ G @ A4 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5690904116761175830ex_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4844_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups7234854612051535045er_int @ G @ A4 )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ B4 ) ) @ ( groups7234854612051535045er_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4845_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups2906978787729119204at_rat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4846_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3539618377306564664at_int @ G @ A4 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3539618377306564664at_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4847_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > uint32] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups8736914816313324502uint32 @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_uint32 @ ( groups8736914816313324502uint32 @ F @ A4 ) @ ( groups8736914816313324502uint32 @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4848_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,F: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
       => ( ( groups8847630953604152069uint32 @ F @ ( minus_2355218937544613996nteger @ A4 @ B4 ) )
          = ( minus_minus_uint32 @ ( groups8847630953604152069uint32 @ F @ A4 ) @ ( groups8847630953604152069uint32 @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4849_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4850_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
       => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4851_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4852_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
       => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A4 ) @ ( groups7234854612051535045er_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4853_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > uint32] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups833757482993574392uint32 @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_uint32 @ ( groups833757482993574392uint32 @ F @ A4 ) @ ( groups833757482993574392uint32 @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4854_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4855_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: int > uint32] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B4 @ A4 )
       => ( ( groups5712668689793887828uint32 @ F @ ( minus_minus_set_int @ A4 @ B4 ) )
          = ( minus_minus_uint32 @ ( groups5712668689793887828uint32 @ F @ A4 ) @ ( groups5712668689793887828uint32 @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4856_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B4 @ A4 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4857_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_4858_sum__diff__nat,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
       => ( ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_4859_sum__diff__nat,axiom,
    ! [B4: set_int,A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ A4 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_4860_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_4861_sum__diff1__nat,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ( member_VEBT_VEBT @ A @ A4 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( minus_minus_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ A @ A4 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( groups771621172384141258BT_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_4862_sum__diff1__nat,axiom,
    ! [A: real,A4: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A4 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A4 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_4863_sum__diff1__nat,axiom,
    ! [A: set_nat,A4: set_set_nat,F: set_nat > nat] :
      ( ( ( member_set_nat @ A @ A4 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_set_nat @ A @ A4 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( groups8294997508430121362at_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_4864_sum__diff1__nat,axiom,
    ! [A: int,A4: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A4 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A4 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_4865_sum__diff1__nat,axiom,
    ! [A: $o,A4: set_o,F: $o > nat] :
      ( ( ( member_o @ A @ A4 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_o @ A @ A4 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
          = ( groups8507830703676809646_o_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_4866_sum__diff1__nat,axiom,
    ! [A: nat,A4: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_4867_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
       => ( ! [B3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( groups2240296850493347238T_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4868_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4869_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4870_sum__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4871_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
       => ( ! [B3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ ( groups136491112297645522BT_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4872_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4873_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4874_sum__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B4 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4875_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4876_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B4 )
       => ( ! [B3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ B4 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) @ ( groups771621172384141258BT_nat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4877_sum_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ( groups2240296850493347238T_real @ G @ A4 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4878_sum_Oremove,axiom,
    ! [A4: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( groups8097168146408367636l_real @ G @ A4 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4879_sum_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ A4 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4880_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( groups1270011288395367621r_real @ G @ A4 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4881_sum_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ( groups136491112297645522BT_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4882_sum_Oremove,axiom,
    ! [A4: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( groups1300246762558778688al_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4883_sum_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( groups5058264527183730370ex_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4884_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( groups6602215022474089585er_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4885_sum_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ( groups771621172384141258BT_nat @ G @ A4 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4886_sum_Oremove,axiom,
    ! [A4: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( groups1935376822645274424al_nat @ G @ A4 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_4887_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N @ ( 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_4888_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).

% cnt_non_neg
thf(fact_4889_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_4890_Bolzano,axiom,
    ! [A: real,B: real,P2: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A3: real,B3: real,C: real] :
            ( ( P2 @ A3 @ B3 )
           => ( ( P2 @ B3 @ C )
             => ( ( ord_less_eq_real @ A3 @ B3 )
               => ( ( ord_less_eq_real @ B3 @ C )
                 => ( P2 @ A3 @ C ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B )
               => ? [D3: real] :
                    ( ( ord_less_real @ zero_zero_real @ D3 )
                    & ! [A3: real,B3: real] :
                        ( ( ( ord_less_eq_real @ A3 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B3 )
                          & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D3 ) )
                       => ( P2 @ A3 @ B3 ) ) ) ) )
         => ( P2 @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_4891_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_4892_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_4893_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_4894_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_4895_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_4896_max__bot,axiom,
    ! [X: assn] :
      ( ( ord_max_assn @ bot_bot_assn @ X )
      = X ) ).

% max_bot
thf(fact_4897_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_4898_max__bot,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X )
      = X ) ).

% max_bot
thf(fact_4899_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_4900_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_4901_max__bot2,axiom,
    ! [X: assn] :
      ( ( ord_max_assn @ X @ bot_bot_assn )
      = X ) ).

% max_bot2
thf(fact_4902_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_4903_max__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ X @ bot_bot_set_o )
      = X ) ).

% max_bot2
thf(fact_4904_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_4905_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_4906_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_4907_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_4908_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_4909_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_4910_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_4911_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_4912_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_4913_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_4914_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_4915_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_4916_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_4917_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_4918_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_4919_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_4920_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_4921_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_4922_Max__insert,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A4 ) )
          = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A4 ) ) ) ) ) ).

% Max_insert
thf(fact_4923_Max__insert,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A4 ) )
          = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A4 ) ) ) ) ) ).

% Max_insert
thf(fact_4924_Max__insert,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A4 ) )
          = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ) ).

% Max_insert
thf(fact_4925_Max__insert,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A4 ) )
          = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ) ).

% Max_insert
thf(fact_4926_max__def,axiom,
    ( ord_max_Code_integer
    = ( ^ [A7: code_integer,B7: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4927_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A7: set_int,B7: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4928_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A7: rat,B7: rat] : ( if_rat @ ( ord_less_eq_rat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4929_max__def,axiom,
    ( ord_max_num
    = ( ^ [A7: num,B7: num] : ( if_num @ ( ord_less_eq_num @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4930_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A7: nat,B7: nat] : ( if_nat @ ( ord_less_eq_nat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4931_max__def,axiom,
    ( ord_max_int
    = ( ^ [A7: int,B7: int] : ( if_int @ ( ord_less_eq_int @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4932_max__absorb1,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y @ X )
     => ( ( ord_max_Code_integer @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4933_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_max_set_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4934_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_max_rat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4935_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_max_num @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4936_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4937_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4938_max__absorb2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_max_Code_integer @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4939_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_max_set_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4940_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_max_rat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4941_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_max_num @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4942_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4943_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4944_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4945_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4946_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4947_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z ) )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4948_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4949_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4950_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4951_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4952_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4953_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4954_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z ) @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_4955_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_4956_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_4957_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_4958_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_4959_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_4960_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_4961_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_4962_Max_Oin__idem,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( ord_max_real @ X @ ( lattic4275903605611617917x_real @ A4 ) )
          = ( lattic4275903605611617917x_real @ A4 ) ) ) ) ).

% Max.in_idem
thf(fact_4963_Max_Oin__idem,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A4 ) )
          = ( lattic4901227151466704046nteger @ A4 ) ) ) ) ).

% Max.in_idem
thf(fact_4964_Max_Oin__idem,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) )
          = ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ).

% Max.in_idem
thf(fact_4965_Max_Oin__idem,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) )
          = ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).

% Max.in_idem
thf(fact_4966_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_4967_hom__Max__commute,axiom,
    ! [H2: $o > $o,N8: set_o] :
      ( ! [X3: $o,Y3: $o] :
          ( ( H2 @ ( ord_max_o @ X3 @ Y3 ) )
          = ( ord_max_o @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_o @ N8 )
       => ( ( N8 != bot_bot_set_o )
         => ( ( H2 @ ( lattic1921953407002678535_Max_o @ N8 ) )
            = ( lattic1921953407002678535_Max_o @ ( image_o_o @ H2 @ N8 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_4968_hom__Max__commute,axiom,
    ! [H2: code_integer > code_integer,N8: set_Code_integer] :
      ( ! [X3: code_integer,Y3: code_integer] :
          ( ( H2 @ ( ord_max_Code_integer @ X3 @ Y3 ) )
          = ( ord_max_Code_integer @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite6017078050557962740nteger @ N8 )
       => ( ( N8 != bot_bo3990330152332043303nteger )
         => ( ( H2 @ ( lattic4901227151466704046nteger @ N8 ) )
            = ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ H2 @ N8 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_4969_hom__Max__commute,axiom,
    ! [H2: int > int,N8: set_int] :
      ( ! [X3: int,Y3: int] :
          ( ( H2 @ ( ord_max_int @ X3 @ Y3 ) )
          = ( ord_max_int @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_int @ N8 )
       => ( ( N8 != bot_bot_set_int )
         => ( ( H2 @ ( lattic8263393255366662781ax_int @ N8 ) )
            = ( lattic8263393255366662781ax_int @ ( image_int_int @ H2 @ N8 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_4970_hom__Max__commute,axiom,
    ! [H2: nat > nat,N8: set_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( H2 @ ( ord_max_nat @ X3 @ Y3 ) )
          = ( ord_max_nat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_nat @ N8 )
       => ( ( N8 != bot_bot_set_nat )
         => ( ( H2 @ ( lattic8265883725875713057ax_nat @ N8 ) )
            = ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ H2 @ N8 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_4971_Max_Osubset,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( B4 != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B4 @ A4 )
         => ( ( ord_max_o @ ( lattic1921953407002678535_Max_o @ B4 ) @ ( lattic1921953407002678535_Max_o @ A4 ) )
            = ( lattic1921953407002678535_Max_o @ A4 ) ) ) ) ) ).

% Max.subset
thf(fact_4972_Max_Osubset,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( B4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le7084787975880047091nteger @ B4 @ A4 )
         => ( ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ B4 ) @ ( lattic4901227151466704046nteger @ A4 ) )
            = ( lattic4901227151466704046nteger @ A4 ) ) ) ) ) ).

% Max.subset
thf(fact_4973_Max_Osubset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( B4 != bot_bot_set_int )
       => ( ( ord_less_eq_set_int @ B4 @ A4 )
         => ( ( ord_max_int @ ( lattic8263393255366662781ax_int @ B4 ) @ ( lattic8263393255366662781ax_int @ A4 ) )
            = ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ) ).

% Max.subset
thf(fact_4974_Max_Osubset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( B4 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B4 @ A4 )
         => ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B4 ) @ ( lattic8265883725875713057ax_nat @ A4 ) )
            = ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ) ).

% Max.subset
thf(fact_4975_Max_Oclosed,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real,Y3: real] : ( member_real @ ( ord_max_real @ X3 @ Y3 ) @ ( insert_real @ X3 @ ( insert_real @ Y3 @ bot_bot_set_real ) ) )
         => ( member_real @ ( lattic4275903605611617917x_real @ A4 ) @ A4 ) ) ) ) ).

% Max.closed
thf(fact_4976_Max_Oclosed,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o,Y3: $o] : ( member_o @ ( ord_max_o @ X3 @ Y3 ) @ ( insert_o @ X3 @ ( insert_o @ Y3 @ bot_bot_set_o ) ) )
         => ( member_o @ ( lattic1921953407002678535_Max_o @ A4 ) @ A4 ) ) ) ) ).

% Max.closed
thf(fact_4977_Max_Oclosed,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,Y3: code_integer] : ( member_Code_integer @ ( ord_max_Code_integer @ X3 @ Y3 ) @ ( insert_Code_integer @ X3 @ ( insert_Code_integer @ Y3 @ bot_bo3990330152332043303nteger ) ) )
         => ( member_Code_integer @ ( lattic4901227151466704046nteger @ A4 ) @ A4 ) ) ) ) ).

% Max.closed
thf(fact_4978_Max_Oclosed,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int,Y3: int] : ( member_int @ ( ord_max_int @ X3 @ Y3 ) @ ( insert_int @ X3 @ ( insert_int @ Y3 @ bot_bot_set_int ) ) )
         => ( member_int @ ( lattic8263393255366662781ax_int @ A4 ) @ A4 ) ) ) ) ).

% Max.closed
thf(fact_4979_Max_Oclosed,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X3: nat,Y3: nat] : ( member_nat @ ( ord_max_nat @ X3 @ Y3 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
         => ( member_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ A4 ) ) ) ) ).

% Max.closed
thf(fact_4980_Max_Oinsert__not__elem,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( A4 != bot_bot_set_real )
         => ( ( lattic4275903605611617917x_real @ ( insert_real @ X @ A4 ) )
            = ( ord_max_real @ X @ ( lattic4275903605611617917x_real @ A4 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_4981_Max_Oinsert__not__elem,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( A4 != bot_bot_set_o )
         => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A4 ) )
            = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A4 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_4982_Max_Oinsert__not__elem,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ~ ( member_Code_integer @ X @ A4 )
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A4 ) )
            = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A4 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_4983_Max_Oinsert__not__elem,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( A4 != bot_bot_set_int )
         => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A4 ) )
            = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_4984_Max_Oinsert__not__elem,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( A4 != bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A4 ) )
            = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_4985_Max_Oinsert__remove,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A4 ) )
        = ( ( ( ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_4986_Max_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
            = bot_bo3990330152332043303nteger )
         => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A4 ) )
            = X ) )
        & ( ( ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
           != bot_bo3990330152332043303nteger )
         => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A4 ) )
            = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_4987_Max_Oinsert__remove,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) )
            = bot_bot_set_int )
         => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A4 ) )
            = X ) )
        & ( ( ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) )
           != bot_bot_set_int )
         => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A4 ) )
            = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_4988_Max_Oinsert__remove,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A4 ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A4 ) )
            = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_4989_Max_Oremove,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( ( ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) )
              = bot_bot_set_real )
           => ( ( lattic4275903605611617917x_real @ A4 )
              = X ) )
          & ( ( ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) )
             != bot_bot_set_real )
           => ( ( lattic4275903605611617917x_real @ A4 )
              = ( ord_max_real @ X @ ( lattic4275903605611617917x_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_4990_Max_Oremove,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ( lattic1921953407002678535_Max_o @ A4 )
          = ( ( ( ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_4991_Max_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( ( ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
              = bot_bo3990330152332043303nteger )
           => ( ( lattic4901227151466704046nteger @ A4 )
              = X ) )
          & ( ( ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
             != bot_bo3990330152332043303nteger )
           => ( ( lattic4901227151466704046nteger @ A4 )
              = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_4992_Max_Oremove,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( ( ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) )
              = bot_bot_set_int )
           => ( ( lattic8263393255366662781ax_int @ A4 )
              = X ) )
          & ( ( ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) )
             != bot_bot_set_int )
           => ( ( lattic8263393255366662781ax_int @ A4 )
              = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_4993_Max_Oremove,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ A4 )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ A4 )
              = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_4994_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4995_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4996_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4997_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4998_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4999_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5000_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5001_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5002_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5003_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5004_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5005_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5006_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X @ Z )
        & ( ord_le6747313008572928689nteger @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_5007_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ( ord_less_real @ X @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_5008_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_5009_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_5010_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_5011_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_5012_max_Obounded__iff,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C2 ) @ A )
      = ( ( ord_le3102999989581377725nteger @ B @ A )
        & ( ord_le3102999989581377725nteger @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_5013_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_5014_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_5015_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_5016_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_5017_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_5018_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_5019_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_5020_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_5021_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_5022_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_5023_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_5024_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_5025_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_5026_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_5027_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5028_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X2: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5029_option_Osize__gen_I1_J,axiom,
    ! [X: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_5030_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_5031_max_Omono,axiom,
    ! [C2: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ A )
     => ( ( ord_le3102999989581377725nteger @ D @ B )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ C2 @ D ) @ ( ord_max_Code_integer @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5032_max_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C2 @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5033_max_Omono,axiom,
    ! [C2: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C2 @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C2 @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5034_max_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C2 @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5035_max_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C2 @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5036_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( A
        = ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.orderE
thf(fact_5037_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_5038_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_5039_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_5040_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_5041_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( ord_max_Code_integer @ A @ B ) )
     => ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% max.orderI
thf(fact_5042_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_5043_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_5044_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_5045_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_5046_max_OboundedE,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C2 ) @ A )
     => ~ ( ( ord_le3102999989581377725nteger @ B @ A )
         => ~ ( ord_le3102999989581377725nteger @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_5047_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_5048_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_5049_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_5050_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_5051_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C2 @ A )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_5052_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_5053_max_OboundedI,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_5054_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_5055_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_5056_max_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B7: code_integer,A7: code_integer] :
          ( A7
          = ( ord_max_Code_integer @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_5057_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A7: rat] :
          ( A7
          = ( ord_max_rat @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_5058_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A7: num] :
          ( A7
          = ( ord_max_num @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_5059_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A7: nat] :
          ( A7
          = ( ord_max_nat @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_5060_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A7: int] :
          ( A7
          = ( ord_max_int @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_5061_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded1
thf(fact_5062_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_5063_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_5064_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_5065_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_5066_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : ( ord_le3102999989581377725nteger @ B @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded2
thf(fact_5067_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_5068_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_5069_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_5070_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_5071_le__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le3102999989581377725nteger @ Z @ X )
        | ( ord_le3102999989581377725nteger @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5072_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5073_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5074_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5075_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5076_max_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B7: code_integer,A7: code_integer] :
          ( ( ord_max_Code_integer @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_5077_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_max_rat @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_5078_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_max_num @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_5079_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_max_nat @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_5080_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_max_int @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_5081_max_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A7: code_integer,B7: code_integer] :
          ( ( ord_max_Code_integer @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_5082_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_max_rat @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_5083_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_max_num @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_5084_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_max_nat @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_5085_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_max_int @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_5086_max_OcoboundedI1,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ A )
     => ( ord_le3102999989581377725nteger @ C2 @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5087_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5088_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C2 @ A )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5089_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5090_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5091_max_OcoboundedI2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ B )
     => ( ord_le3102999989581377725nteger @ C2 @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5092_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ B )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5093_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C2 @ B )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5094_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5095_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ B )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5096_max_Ostrict__coboundedI2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ B )
     => ( ord_le6747313008572928689nteger @ C2 @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5097_max_Ostrict__coboundedI2,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ C2 @ B )
     => ( ord_less_real @ C2 @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5098_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ B )
     => ( ord_less_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5099_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( ( ord_less_num @ C2 @ B )
     => ( ord_less_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5100_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5101_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_int @ C2 @ B )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5102_max_Ostrict__coboundedI1,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ A )
     => ( ord_le6747313008572928689nteger @ C2 @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5103_max_Ostrict__coboundedI1,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ A )
     => ( ord_less_real @ C2 @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5104_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ A )
     => ( ord_less_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5105_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( ( ord_less_num @ C2 @ A )
     => ( ord_less_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5106_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5107_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ A )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5108_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B7: code_integer,A7: code_integer] :
          ( ( A7
            = ( ord_max_Code_integer @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5109_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A7: real] :
          ( ( A7
            = ( ord_max_real @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5110_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( A7
            = ( ord_max_rat @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5111_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A7: num] :
          ( ( A7
            = ( ord_max_num @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5112_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( A7
            = ( ord_max_nat @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5113_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A7: int] :
          ( ( A7
            = ( ord_max_int @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5114_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B @ C2 ) @ A )
     => ~ ( ( ord_le6747313008572928689nteger @ B @ A )
         => ~ ( ord_le6747313008572928689nteger @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5115_max_Ostrict__boundedE,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C2 ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5116_max_Ostrict__boundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5117_max_Ostrict__boundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C2 ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5118_max_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5119_max_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5120_less__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X )
        | ( ord_le6747313008572928689nteger @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5121_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z @ X )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5122_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5123_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5124_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5125_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5126_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_o,X: $o] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ L ) )
         => ( ( sorted_wrt_o @ ord_less_eq_o @ L )
           => ( ( ord_less_eq_o @ ( nth_o @ L @ I ) @ X )
             => ( ( ord_less_o @ X @ ( nth_o @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_o @ ( nth_o @ L @ K2 ) @ X )
                         => ~ ( ord_less_o @ X @ ( nth_o @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5127_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_real,X: real] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_real @ L ) )
         => ( ( sorted_wrt_real @ ord_less_eq_real @ L )
           => ( ( ord_less_eq_real @ ( nth_real @ L @ I ) @ X )
             => ( ( ord_less_real @ X @ ( nth_real @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_real @ ( nth_real @ L @ K2 ) @ X )
                         => ~ ( ord_less_real @ X @ ( nth_real @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5128_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_rat,X: rat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_rat @ L ) )
         => ( ( sorted_wrt_rat @ ord_less_eq_rat @ L )
           => ( ( ord_less_eq_rat @ ( nth_rat @ L @ I ) @ X )
             => ( ( ord_less_rat @ X @ ( nth_rat @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_rat @ ( nth_rat @ L @ K2 ) @ X )
                         => ~ ( ord_less_rat @ X @ ( nth_rat @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5129_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_num,X: num] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_num @ L ) )
         => ( ( sorted_wrt_num @ ord_less_eq_num @ L )
           => ( ( ord_less_eq_num @ ( nth_num @ L @ I ) @ X )
             => ( ( ord_less_num @ X @ ( nth_num @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_num @ ( nth_num @ L @ K2 ) @ X )
                         => ~ ( ord_less_num @ X @ ( nth_num @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5130_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
         => ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
           => ( ( ord_less_eq_nat @ ( nth_nat @ L @ I ) @ X )
             => ( ( ord_less_nat @ X @ ( nth_nat @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_nat @ ( nth_nat @ L @ K2 ) @ X )
                         => ~ ( ord_less_nat @ X @ ( nth_nat @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5131_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_int,X: int] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ L ) )
         => ( ( sorted_wrt_int @ ord_less_eq_int @ L )
           => ( ( ord_less_eq_int @ ( nth_int @ L @ I ) @ X )
             => ( ( ord_less_int @ X @ ( nth_int @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_int @ ( nth_int @ L @ K2 ) @ X )
                         => ~ ( ord_less_int @ X @ ( nth_int @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5132_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBTi,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( nth_Pr3244165891152107629_VEBTi @ ( enumerate_VEBT_VEBTi @ N @ Xs ) @ M )
        = ( produc2649746096677893406_VEBTi @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBTi @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5133_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5134_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5135_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_real,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_real @ Xs ) )
     => ( ( nth_Pr7767817059697521252t_real @ ( enumerate_real @ N @ Xs ) @ M )
        = ( produc7837566107596912789t_real @ ( plus_plus_nat @ N @ M ) @ ( nth_real @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5136_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs ) @ M )
        = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5137_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5138_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5139_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2809031099982602151Ti_nat @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5140_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5141_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2806540629473551875Ti_int @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5142_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L2018189785592951398T_VEBT @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2018189785592951398T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5143_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8891422820522952478nt_nat @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5144_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L114188773329725699_VEBTi @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L114188773329725699_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5145_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8888932350013902202nt_int @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5146_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5147_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ I5 @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5148_nth__step__trancl,axiom,
    ! [Xs: list_VEBT_VEBTi,R2: set_Pr2227491710730465451_VEBTi,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ one_one_nat ) )
         => ( member660371905731732212_VEBTi @ ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ ( suc @ N2 ) ) @ ( nth_VEBT_VEBTi @ Xs @ N2 ) ) @ R2 ) )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member660371905731732212_VEBTi @ ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ N ) @ ( nth_VEBT_VEBTi @ Xs @ M ) ) @ ( transi2803566869205510612_VEBTi @ R2 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5149_nth__step__trancl,axiom,
    ! [Xs: list_uint32,R2: set_Pr1773385645901665561uint32,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_s4844771616002835472uint32 @ Xs ) @ one_one_nat ) )
         => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ ( nth_uint32 @ Xs @ ( suc @ N2 ) ) @ ( nth_uint32 @ Xs @ N2 ) ) @ R2 ) )
     => ( ( ord_less_nat @ N @ ( size_s4844771616002835472uint32 @ Xs ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ ( nth_uint32 @ Xs @ N ) @ ( nth_uint32 @ Xs @ M ) ) @ ( transi3114468042090999947uint32 @ R2 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5150_nth__step__trancl,axiom,
    ! [Xs: list_int,R2: set_Pr958786334691620121nt_int,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ ( suc @ N2 ) ) @ ( nth_int @ Xs @ N2 ) ) @ R2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N ) @ ( nth_int @ Xs @ M ) ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5151_nth__step__trancl,axiom,
    ! [Xs: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat ) )
         => ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( suc @ N2 ) ) @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) @ R2 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) @ ( transi8906537157094044885T_VEBT @ R2 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5152_nth__step__trancl,axiom,
    ! [Xs: list_real,R2: set_Pr6218003697084177305l_real,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_real @ Xs ) @ one_one_nat ) )
         => ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ ( nth_real @ Xs @ ( suc @ N2 ) ) @ ( nth_real @ Xs @ N2 ) ) @ R2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ ( nth_real @ Xs @ N ) @ ( nth_real @ Xs @ M ) ) @ ( transi1789104906590519371l_real @ R2 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5153_nth__step__trancl,axiom,
    ! [Xs: list_o,R2: set_Product_prod_o_o,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_o @ Xs ) @ one_one_nat ) )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ ( suc @ N2 ) ) @ ( nth_o @ Xs @ N2 ) ) @ R2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N ) @ ( nth_o @ Xs @ M ) ) @ ( transitive_trancl_o @ R2 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5154_nth__step__trancl,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( suc @ N2 ) ) @ ( nth_nat @ Xs @ N2 ) ) @ R2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N ) @ ( nth_nat @ Xs @ M ) ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5155_nth__Cons__pos,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
        = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5156_nth__Cons__pos,axiom,
    ! [N: nat,X: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ N )
        = ( nth_VEBT_VEBTi @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5157_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5158_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5159_nth__Cons__pos,axiom,
    ! [N: nat,X: $o,Xs: list_o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_o @ ( cons_o @ X @ Xs ) @ N )
        = ( nth_o @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5160_length__nth__simps_I4_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBT @ Xs @ N ) ) ).

% length_nth_simps(4)
thf(fact_5161_length__nth__simps_I4_J,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBTi @ Xs @ N ) ) ).

% length_nth_simps(4)
thf(fact_5162_length__nth__simps_I4_J,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_int @ Xs @ N ) ) ).

% length_nth_simps(4)
thf(fact_5163_length__nth__simps_I4_J,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% length_nth_simps(4)
thf(fact_5164_length__nth__simps_I4_J,axiom,
    ! [X: $o,Xs: list_o,N: nat] :
      ( ( nth_o @ ( cons_o @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_o @ Xs @ N ) ) ).

% length_nth_simps(4)
thf(fact_5165_nth__Cons__Suc,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBT @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_5166_nth__Cons__Suc,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBTi @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_5167_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_int @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_5168_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_5169_nth__Cons__Suc,axiom,
    ! [X: $o,Xs: list_o,N: nat] :
      ( ( nth_o @ ( cons_o @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_o @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_5170_length__nth__simps_I3_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% length_nth_simps(3)
thf(fact_5171_length__nth__simps_I3_J,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% length_nth_simps(3)
thf(fact_5172_length__nth__simps_I3_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% length_nth_simps(3)
thf(fact_5173_length__nth__simps_I3_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% length_nth_simps(3)
thf(fact_5174_length__nth__simps_I3_J,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( nth_o @ ( cons_o @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% length_nth_simps(3)
thf(fact_5175_nth__Cons__0,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5176_nth__Cons__0,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5177_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5178_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5179_nth__Cons__0,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( nth_o @ ( cons_o @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5180_trancl__single,axiom,
    ! [A: uint32,B: uint32] :
      ( ( transi3114468042090999947uint32 @ ( insert4454361187789264009uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ bot_bo8438649754162204037uint32 ) )
      = ( insert4454361187789264009uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ bot_bo8438649754162204037uint32 ) ) ).

% trancl_single
thf(fact_5181_trancl__single,axiom,
    ! [A: nat,B: nat] :
      ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ bot_bo2099793752762293965at_nat ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ bot_bo2099793752762293965at_nat ) ) ).

% trancl_single
thf(fact_5182_trancl__single,axiom,
    ! [A: int,B: int] :
      ( ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B ) @ bot_bo1796632182523588997nt_int ) )
      = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B ) @ bot_bo1796632182523588997nt_int ) ) ).

% trancl_single
thf(fact_5183_listI__assn__finite,axiom,
    ! [I5: set_nat,A4: vEBT_VEBT > vEBT_VEBTi > assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] :
      ( ~ ( finite_finite_nat @ I5 )
     => ( ( vEBT_L1528199826722428489_VEBTi @ I5 @ A4 @ Xs @ Xsi )
        = bot_bot_assn ) ) ).

% listI_assn_finite
thf(fact_5184_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( enumerate_int @ N @ ( cons_int @ X @ Xs ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_5185_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: $o,Xs: list_o] :
      ( ( enumerate_o @ N @ ( cons_o @ X @ Xs ) )
      = ( cons_P9142372351690779143_nat_o @ ( product_Pair_nat_o @ N @ X ) @ ( enumerate_o @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_5186_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_5187_list__tail__coinc,axiom,
    ! [N1: int,R1: list_int,N22: int,R22: list_int] :
      ( ( ( cons_int @ N1 @ R1 )
        = ( cons_int @ N22 @ R22 ) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
thf(fact_5188_list__tail__coinc,axiom,
    ! [N1: nat,R1: list_nat,N22: nat,R22: list_nat] :
      ( ( ( cons_nat @ N1 @ R1 )
        = ( cons_nat @ N22 @ R22 ) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
thf(fact_5189_list__tail__coinc,axiom,
    ! [N1: $o,R1: list_o,N22: $o,R22: list_o] :
      ( ( ( cons_o @ N1 @ R1 )
        = ( cons_o @ N22 @ R22 ) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
thf(fact_5190_sorted2,axiom,
    ! [X: $o,Y: $o,Zs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( cons_o @ X @ ( cons_o @ Y @ Zs ) ) )
      = ( ( ord_less_eq_o @ X @ Y )
        & ( sorted_wrt_o @ ord_less_eq_o @ ( cons_o @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5191_sorted2,axiom,
    ! [X: rat,Y: rat,Zs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ X @ ( cons_rat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_rat @ X @ Y )
        & ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5192_sorted2,axiom,
    ! [X: num,Y: num,Zs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ X @ ( cons_num @ Y @ Zs ) ) )
      = ( ( ord_less_eq_num @ X @ Y )
        & ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5193_sorted2,axiom,
    ! [X: nat,Y: nat,Zs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5194_sorted2,axiom,
    ! [X: int,Y: int,Zs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X @ ( cons_int @ Y @ Zs ) ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5195_sorted__simps_I2_J,axiom,
    ! [X: real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ ( cons_real @ X @ Ys ) )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Ys ) )
           => ( ord_less_eq_real @ X @ X4 ) )
        & ( sorted_wrt_real @ ord_less_eq_real @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_5196_sorted__simps_I2_J,axiom,
    ! [X: $o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( cons_o @ X @ Ys ) )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Ys ) )
           => ( ord_less_eq_o @ X @ X4 ) )
        & ( sorted_wrt_o @ ord_less_eq_o @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_5197_sorted__simps_I2_J,axiom,
    ! [X: rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ X @ Ys ) )
      = ( ! [X4: rat] :
            ( ( member_rat @ X4 @ ( set_rat2 @ Ys ) )
           => ( ord_less_eq_rat @ X @ X4 ) )
        & ( sorted_wrt_rat @ ord_less_eq_rat @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_5198_sorted__simps_I2_J,axiom,
    ! [X: num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ X @ Ys ) )
      = ( ! [X4: num] :
            ( ( member_num @ X4 @ ( set_num2 @ Ys ) )
           => ( ord_less_eq_num @ X @ X4 ) )
        & ( sorted_wrt_num @ ord_less_eq_num @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_5199_sorted__simps_I2_J,axiom,
    ! [X: nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ord_less_eq_nat @ X @ X4 ) )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_5200_sorted__simps_I2_J,axiom,
    ! [X: int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X @ Ys ) )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Ys ) )
           => ( ord_less_eq_int @ X @ X4 ) )
        & ( sorted_wrt_int @ ord_less_eq_int @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_5201_strict__sorted__simps_I2_J,axiom,
    ! [X: $o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_o @ ( cons_o @ X @ Ys ) )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Ys ) )
           => ( ord_less_o @ X @ X4 ) )
        & ( sorted_wrt_o @ ord_less_o @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_5202_strict__sorted__simps_I2_J,axiom,
    ! [X: real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_real @ ( cons_real @ X @ Ys ) )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Ys ) )
           => ( ord_less_real @ X @ X4 ) )
        & ( sorted_wrt_real @ ord_less_real @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_5203_strict__sorted__simps_I2_J,axiom,
    ! [X: rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ ( cons_rat @ X @ Ys ) )
      = ( ! [X4: rat] :
            ( ( member_rat @ X4 @ ( set_rat2 @ Ys ) )
           => ( ord_less_rat @ X @ X4 ) )
        & ( sorted_wrt_rat @ ord_less_rat @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_5204_strict__sorted__simps_I2_J,axiom,
    ! [X: num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ ( cons_num @ X @ Ys ) )
      = ( ! [X4: num] :
            ( ( member_num @ X4 @ ( set_num2 @ Ys ) )
           => ( ord_less_num @ X @ X4 ) )
        & ( sorted_wrt_num @ ord_less_num @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_5205_strict__sorted__simps_I2_J,axiom,
    ! [X: nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X @ Ys ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ord_less_nat @ X @ X4 ) )
        & ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_5206_strict__sorted__simps_I2_J,axiom,
    ! [X: int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ ( cons_int @ X @ Ys ) )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Ys ) )
           => ( ord_less_int @ X @ X4 ) )
        & ( sorted_wrt_int @ ord_less_int @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_5207_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_real] :
      ( ( sorted_wrt_real @ ord_less_real @ Xs )
     => ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_5208_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ Xs )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_5209_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ Xs )
     => ( sorted_wrt_num @ ord_less_eq_num @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_5210_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_5211_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_5212_strict__sorted__equal,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_o @ Xs )
     => ( ( sorted_wrt_o @ ord_less_o @ Ys )
       => ( ( ( set_o2 @ Ys )
            = ( set_o2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_5213_strict__sorted__equal,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_real @ Xs )
     => ( ( sorted_wrt_real @ ord_less_real @ Ys )
       => ( ( ( set_real2 @ Ys )
            = ( set_real2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_5214_strict__sorted__equal,axiom,
    ! [Xs: list_rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ Xs )
     => ( ( sorted_wrt_rat @ ord_less_rat @ Ys )
       => ( ( ( set_rat2 @ Ys )
            = ( set_rat2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_5215_strict__sorted__equal,axiom,
    ! [Xs: list_num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ Xs )
     => ( ( sorted_wrt_num @ ord_less_num @ Ys )
       => ( ( ( set_num2 @ Ys )
            = ( set_num2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_5216_strict__sorted__equal,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
       => ( ( ( set_nat2 @ Ys )
            = ( set_nat2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_5217_strict__sorted__equal,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs )
     => ( ( sorted_wrt_int @ ord_less_int @ Ys )
       => ( ( ( set_int2 @ Ys )
            = ( set_int2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_5218_length__nth__simps_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
      = ( suc @ ( size_size_list_int @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_5219_length__nth__simps_I2_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) )
      = ( suc @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_5220_length__nth__simps_I2_J,axiom,
    ! [X: real,Xs: list_real] :
      ( ( size_size_list_real @ ( cons_real @ X @ Xs ) )
      = ( suc @ ( size_size_list_real @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_5221_length__nth__simps_I2_J,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X @ Xs ) )
      = ( suc @ ( size_size_list_o @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_5222_length__nth__simps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_5223_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y4 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_5224_length__Suc__conv,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs
              = ( cons_VEBT_VEBT @ Y4 @ Ys3 ) )
            & ( ( size_s6755466524823107622T_VEBT @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_5225_length__Suc__conv,axiom,
    ! [Xs: list_real,N: nat] :
      ( ( ( size_size_list_real @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: real,Ys3: list_real] :
            ( ( Xs
              = ( cons_real @ Y4 @ Ys3 ) )
            & ( ( size_size_list_real @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_5226_length__Suc__conv,axiom,
    ! [Xs: list_o,N: nat] :
      ( ( ( size_size_list_o @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: $o,Ys3: list_o] :
            ( ( Xs
              = ( cons_o @ Y4 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_5227_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_5228_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ( suc @ N )
        = ( size_size_list_int @ Xs ) )
      = ( ? [Y4: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y4 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_5229_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ( suc @ N )
        = ( size_s6755466524823107622T_VEBT @ Xs ) )
      = ( ? [Y4: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs
              = ( cons_VEBT_VEBT @ Y4 @ Ys3 ) )
            & ( ( size_s6755466524823107622T_VEBT @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_5230_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ( suc @ N )
        = ( size_size_list_real @ Xs ) )
      = ( ? [Y4: real,Ys3: list_real] :
            ( ( Xs
              = ( cons_real @ Y4 @ Ys3 ) )
            & ( ( size_size_list_real @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_5231_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ( suc @ N )
        = ( size_size_list_o @ Xs ) )
      = ( ? [Y4: $o,Ys3: list_o] :
            ( ( Xs
              = ( cons_o @ Y4 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_5232_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_5233_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
     => ( Xs
       != ( cons_int @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_5234_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs
       != ( cons_VEBT_VEBT @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_5235_impossible__Cons,axiom,
    ! [Xs: list_real,Ys: list_real,X: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ ( size_size_list_real @ Ys ) )
     => ( Xs
       != ( cons_real @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_5236_impossible__Cons,axiom,
    ! [Xs: list_o,Ys: list_o,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) )
     => ( Xs
       != ( cons_o @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_5237_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_5238_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5239_set__subset__Cons,axiom,
    ! [Xs: list_real,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ ( set_real2 @ ( cons_real @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5240_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5241_set__subset__Cons,axiom,
    ! [Xs: list_o,X: $o] : ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ ( set_o2 @ ( cons_o @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5242_set__subset__Cons,axiom,
    ! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5243_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBTi > assn,A9: vEBT_VEBTi > vEBT_VEBTi > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
            = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_VEBT_VEBTi @ Xsi @ I2 )
                        = ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) @ ( nth_VEBT_VEBTi @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L886525131989349516_VEBTi @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5244_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int,A4: vEBT_VEBTi > int > assn,A9: vEBT_VEBTi > int > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_size_list_int @ Xsi )
            = ( size_size_list_int @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_size_list_int @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_int @ Xsi @ I2 )
                        = ( nth_int @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) @ ( nth_int @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_int @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L2806540629473551875Ti_int @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L2806540629473551875Ti_int @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5245_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A4: int > vEBT_VEBTi > assn,A9: int > vEBT_VEBTi > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Xs4 ) )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
            = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                 => ( ( ( size_size_list_int @ Xs )
                      = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                   => ( ( ( nth_int @ Xs @ I2 )
                        = ( nth_int @ Xs4 @ I2 ) )
                      & ( ( nth_VEBT_VEBTi @ Xsi @ I2 )
                        = ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_int @ Xs @ I2 ) @ ( nth_VEBT_VEBTi @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_int @ Xs4 @ I2 ) @ ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L114188773329725699_VEBTi @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L114188773329725699_VEBTi @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5246_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int,A4: int > int > assn,A9: int > int > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Xs4 ) )
       => ( ( ( size_size_list_int @ Xsi )
            = ( size_size_list_int @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                 => ( ( ( size_size_list_int @ Xs )
                      = ( size_size_list_int @ Xsi ) )
                   => ( ( ( nth_int @ Xs @ I2 )
                        = ( nth_int @ Xs4 @ I2 ) )
                      & ( ( nth_int @ Xsi @ I2 )
                        = ( nth_int @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_int @ Xs4 @ I2 ) @ ( nth_int @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L8888932350013902202nt_int @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L8888932350013902202nt_int @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5247_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A4: vEBT_VEBTi > vEBT_VEBT > assn,A9: vEBT_VEBTi > vEBT_VEBT > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) @ ( nth_VEBT_VEBT @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L2497118539674116125T_VEBT @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5248_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A4: int > vEBT_VEBT > assn,A9: int > vEBT_VEBT > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                 => ( ( ( size_size_list_int @ Xs )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_int @ Xs @ I2 )
                        = ( nth_int @ Xs4 @ I2 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_int @ Xs @ I2 ) @ ( nth_VEBT_VEBT @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_int @ Xs4 @ I2 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L2018189785592951398T_VEBT @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L2018189785592951398T_VEBT @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5249_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real,A4: vEBT_VEBTi > real > assn,A9: vEBT_VEBTi > real > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_size_list_real @ Xsi )
            = ( size_size_list_real @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_real @ Xsi @ I2 )
                        = ( nth_real @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) @ ( nth_real @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_real @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L7728200936804140803i_real @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L7728200936804140803i_real @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5250_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_real,Xsi2: list_real,A4: int > real > assn,A9: int > real > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Xs4 ) )
       => ( ( ( size_size_list_real @ Xsi )
            = ( size_size_list_real @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                 => ( ( ( size_size_list_int @ Xs )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_int @ Xs @ I2 )
                        = ( nth_int @ Xs4 @ I2 ) )
                      & ( ( nth_real @ Xsi @ I2 )
                        = ( nth_real @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_int @ Xs @ I2 ) @ ( nth_real @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_int @ Xs4 @ I2 ) @ ( nth_real @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L7077748017936769786t_real @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L7077748017936769786t_real @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5251_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o,A4: vEBT_VEBTi > $o > assn,A9: vEBT_VEBTi > $o > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_size_list_o @ Xsi )
            = ( size_size_list_o @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_o @ Xsi @ I2 )
                        = ( nth_o @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) @ ( nth_o @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_o @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L3328983362619735041EBTi_o @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L3328983362619735041EBTi_o @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5252_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_o,Xsi2: list_o,A4: int > $o > assn,A9: int > $o > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Xs4 ) )
       => ( ( ( size_size_list_o @ Xsi )
            = ( size_size_list_o @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                 => ( ( ( size_size_list_int @ Xs )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_int @ Xs @ I2 )
                        = ( nth_int @ Xs4 @ I2 ) )
                      & ( ( nth_o @ Xsi @ I2 )
                        = ( nth_o @ Xsi2 @ I2 ) )
                      & ( ( A4 @ ( nth_int @ Xs @ I2 ) @ ( nth_o @ Xsi @ I2 ) )
                        = ( A9 @ ( nth_int @ Xs4 @ I2 ) @ ( nth_o @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L3563379889750563018_int_o @ I5 @ A4 @ Xs @ Xsi )
              = ( vEBT_L3563379889750563018_int_o @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5253_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: vEBT_VEBTi > vEBT_VEBTi > assn,A9: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
              = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_VEBT_VEBTi @ Xsi @ I2 )
                          = ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L886525131989349516_VEBTi @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5254_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: vEBT_VEBTi > int > assn,A9: vEBT_VEBTi > int > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_size_list_int @ Xsi )
              = ( size_size_list_int @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_size_list_int @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_int @ Xsi @ I2 )
                          = ( nth_int @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L2806540629473551875Ti_int @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L2806540629473551875Ti_int @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5255_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: int > vEBT_VEBTi > assn,A9: int > vEBT_VEBTi > assn,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Xs4 ) )
         => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
              = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                   => ( ( ( size_size_list_int @ Xs )
                        = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                     => ( ( ( nth_int @ Xs @ I2 )
                          = ( nth_int @ Xs4 @ I2 ) )
                        & ( ( nth_VEBT_VEBTi @ Xsi @ I2 )
                          = ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L114188773329725699_VEBTi @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L114188773329725699_VEBTi @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5256_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: int > int > assn,A9: int > int > assn,Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Xs4 ) )
         => ( ( ( size_size_list_int @ Xsi )
              = ( size_size_list_int @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                   => ( ( ( size_size_list_int @ Xs )
                        = ( size_size_list_int @ Xsi ) )
                     => ( ( ( nth_int @ Xs @ I2 )
                          = ( nth_int @ Xs4 @ I2 ) )
                        & ( ( nth_int @ Xsi @ I2 )
                          = ( nth_int @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L8888932350013902202nt_int @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L8888932350013902202nt_int @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5257_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: vEBT_VEBTi > vEBT_VEBT > assn,A9: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L2497118539674116125T_VEBT @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5258_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: int > vEBT_VEBT > assn,A9: int > vEBT_VEBT > assn,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                   => ( ( ( size_size_list_int @ Xs )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_int @ Xs @ I2 )
                          = ( nth_int @ Xs4 @ I2 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L2018189785592951398T_VEBT @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L2018189785592951398T_VEBT @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5259_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: vEBT_VEBTi > real > assn,A9: vEBT_VEBTi > real > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_size_list_real @ Xsi )
              = ( size_size_list_real @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_real @ Xsi @ I2 )
                          = ( nth_real @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L7728200936804140803i_real @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L7728200936804140803i_real @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5260_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: int > real > assn,A9: int > real > assn,Xs: list_int,Xs4: list_int,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Xs4 ) )
         => ( ( ( size_size_list_real @ Xsi )
              = ( size_size_list_real @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                   => ( ( ( size_size_list_int @ Xs )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_int @ Xs @ I2 )
                          = ( nth_int @ Xs4 @ I2 ) )
                        & ( ( nth_real @ Xsi @ I2 )
                          = ( nth_real @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L7077748017936769786t_real @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L7077748017936769786t_real @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5261_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: vEBT_VEBTi > $o > assn,A9: vEBT_VEBTi > $o > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_size_list_o @ Xsi )
              = ( size_size_list_o @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_o @ Xsi @ I2 )
                          = ( nth_o @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L3328983362619735041EBTi_o @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L3328983362619735041EBTi_o @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5262_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: int > $o > assn,A9: int > $o > assn,Xs: list_int,Xs4: list_int,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Xs4 ) )
         => ( ( ( size_size_list_o @ Xsi )
              = ( size_size_list_o @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
                   => ( ( ( size_size_list_int @ Xs )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_int @ Xs @ I2 )
                          = ( nth_int @ Xs4 @ I2 ) )
                        & ( ( nth_o @ Xsi @ I2 )
                          = ( nth_o @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L3563379889750563018_int_o @ I5 @ A4 @ Xs @ Xsi )
                = ( vEBT_L3563379889750563018_int_o @ I6 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5263_sorted__wrt01,axiom,
    ! [Xs: list_int,P2: int > int > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( sorted_wrt_int @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5264_sorted__wrt01,axiom,
    ! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > vEBT_VEBT > $o] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
     => ( sorted_wrt_VEBT_VEBT @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5265_sorted__wrt01,axiom,
    ! [Xs: list_real,P2: real > real > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ one_one_nat )
     => ( sorted_wrt_real @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5266_sorted__wrt01,axiom,
    ! [Xs: list_o,P2: $o > $o > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( sorted_wrt_o @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5267_sorted__wrt01,axiom,
    ! [Xs: list_nat,P2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_nat @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5268_sorted__wrt__iff__nth__less,axiom,
    ( sorted9206477368072086664_VEBTi
    = ( ^ [P5: vEBT_VEBTi > vEBT_VEBTi > $o,Xs2: list_VEBT_VEBTi] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( P5 @ ( nth_VEBT_VEBTi @ Xs2 @ I3 ) @ ( nth_VEBT_VEBTi @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5269_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_int
    = ( ^ [P5: int > int > $o,Xs2: list_int] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
           => ( P5 @ ( nth_int @ Xs2 @ I3 ) @ ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5270_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_VEBT_VEBT
    = ( ^ [P5: vEBT_VEBT > vEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P5 @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) @ ( nth_VEBT_VEBT @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5271_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_real
    = ( ^ [P5: real > real > $o,Xs2: list_real] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs2 ) )
           => ( P5 @ ( nth_real @ Xs2 @ I3 ) @ ( nth_real @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5272_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_o
    = ( ^ [P5: $o > $o > $o,Xs2: list_o] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs2 ) )
           => ( P5 @ ( nth_o @ Xs2 @ I3 ) @ ( nth_o @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5273_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P5: nat > nat > $o,Xs2: list_nat] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
           => ( P5 @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5274_sorted__wrt__nth__less,axiom,
    ! [P2: vEBT_VEBTi > vEBT_VEBTi > $o,Xs: list_VEBT_VEBTi,I: nat,J: nat] :
      ( ( sorted9206477368072086664_VEBTi @ P2 @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs ) )
         => ( P2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5275_sorted__wrt__nth__less,axiom,
    ! [P2: int > int > $o,Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ P2 @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( P2 @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5276_sorted__wrt__nth__less,axiom,
    ! [P2: vEBT_VEBT > vEBT_VEBT > $o,Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( sorted_wrt_VEBT_VEBT @ P2 @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5277_sorted__wrt__nth__less,axiom,
    ! [P2: real > real > $o,Xs: list_real,I: nat,J: nat] :
      ( ( sorted_wrt_real @ P2 @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
         => ( P2 @ ( nth_real @ Xs @ I ) @ ( nth_real @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5278_sorted__wrt__nth__less,axiom,
    ! [P2: $o > $o > $o,Xs: list_o,I: nat,J: nat] :
      ( ( sorted_wrt_o @ P2 @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
         => ( P2 @ ( nth_o @ Xs @ I ) @ ( nth_o @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5279_sorted__wrt__nth__less,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ P2 @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( P2 @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5280_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_5281_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
      = ( ? [X4: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5282_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) )
      = ( ? [X4: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs
              = ( cons_VEBT_VEBT @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5283_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_real @ Xs ) )
      = ( ? [X4: real,Ys3: list_real] :
            ( ( Xs
              = ( cons_real @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_real @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5284_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) )
      = ( ? [X4: $o,Ys3: list_o] :
            ( ( Xs
              = ( cons_o @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5285_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X4: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5286_sorted01,axiom,
    ! [Xs: list_real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ one_one_nat )
     => ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).

% sorted01
thf(fact_5287_sorted01,axiom,
    ! [Xs: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( sorted_wrt_o @ ord_less_eq_o @ Xs ) ) ).

% sorted01
thf(fact_5288_sorted01,axiom,
    ! [Xs: list_rat] :
      ( ( ord_less_eq_nat @ ( size_size_list_rat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ Xs ) ) ).

% sorted01
thf(fact_5289_sorted01,axiom,
    ! [Xs: list_num] :
      ( ( ord_less_eq_nat @ ( size_size_list_num @ Xs ) @ one_one_nat )
     => ( sorted_wrt_num @ ord_less_eq_num @ Xs ) ) ).

% sorted01
thf(fact_5290_sorted01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% sorted01
thf(fact_5291_sorted01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).

% sorted01
thf(fact_5292_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs ) )
             => ( ord_less_eq_real @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5293_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs @ I3 ) @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5294_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_rat @ Xs ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I3 ) @ ( nth_rat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5295_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs @ I3 ) @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5296_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5297_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5298_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5299_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5300_list_Osize_I4_J,axiom,
    ! [X21: real,X22: list_real] :
      ( ( size_size_list_real @ ( cons_real @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_real @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5301_list_Osize_I4_J,axiom,
    ! [X21: $o,X22: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_o @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5302_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5303_nth__Cons_H,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
          = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5304_nth__Cons_H,axiom,
    ! [N: nat,X: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ N )
          = ( nth_VEBT_VEBTi @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5305_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5306_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5307_nth__Cons_H,axiom,
    ! [X: $o,Xs: list_o,N: nat] :
      ( ( nth_o @ ( cons_o @ X @ Xs ) @ N )
      = ( ( ( N = zero_zero_nat )
         => X )
        & ( ( N != zero_zero_nat )
         => ( nth_o @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5308_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5309_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2809031099982602151Ti_nat @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5310_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5311_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2806540629473551875Ti_int @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5312_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L2018189785592951398T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2018189785592951398T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5313_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8891422820522952478nt_nat @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5314_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L114188773329725699_VEBTi @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L114188773329725699_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5315_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: int > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8888932350013902202nt_int @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5316_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5317_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ ( insert_nat @ I @ I5 ) @ A4 @ Xs @ Xsi )
          = ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5318_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_real @ Xs ) )
           => ( ord_less_eq_real @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5319_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_o @ Xs ) )
           => ( ord_less_eq_o @ ( nth_o @ Xs @ I3 ) @ ( nth_o @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5320_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_rat @ Xs ) )
           => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I3 ) @ ( nth_rat @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5321_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_num @ Xs ) )
           => ( ord_less_eq_num @ ( nth_num @ Xs @ I3 ) @ ( nth_num @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5322_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5323_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5324_sorted__iff__nth__mono,axiom,
    ! [Xs: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs ) )
             => ( ord_less_eq_real @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5325_sorted__iff__nth__mono,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs @ I3 ) @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5326_sorted__iff__nth__mono,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_rat @ Xs ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I3 ) @ ( nth_rat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5327_sorted__iff__nth__mono,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs @ I3 ) @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5328_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5329_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5330_sorted__nth__mono,axiom,
    ! [Xs: list_real,I: nat,J: nat] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
         => ( ord_less_eq_real @ ( nth_real @ Xs @ I ) @ ( nth_real @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5331_sorted__nth__mono,axiom,
    ! [Xs: list_o,I: nat,J: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
         => ( ord_less_eq_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5332_sorted__nth__mono,axiom,
    ! [Xs: list_rat,I: nat,J: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_rat @ Xs ) )
         => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I ) @ ( nth_rat @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5333_sorted__nth__mono,axiom,
    ! [Xs: list_num,I: nat,J: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_num @ Xs ) )
         => ( ord_less_eq_num @ ( nth_num @ Xs @ I ) @ ( nth_num @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5334_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5335_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5336_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ~ ! [L2: list_literal] :
            ( ( sorted_wrt_literal @ ord_less_literal @ L2 )
           => ( ( ( set_literal2 @ L2 )
                = A4 )
             => ( ( size_s2501651207091587910iteral @ L2 )
               != ( finite_card_literal @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5337_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ~ ! [L2: list_Code_integer] :
            ( ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ L2 )
           => ( ( ( set_Code_integer2 @ L2 )
                = A4 )
             => ( ( size_s3445333598471063425nteger @ L2 )
               != ( finite4902975817058060853nteger @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5338_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ~ ! [L2: list_o] :
            ( ( sorted_wrt_o @ ord_less_o @ L2 )
           => ( ( ( set_o2 @ L2 )
                = A4 )
             => ( ( size_size_list_o @ L2 )
               != ( finite_card_o @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5339_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ~ ! [L2: list_real] :
            ( ( sorted_wrt_real @ ord_less_real @ L2 )
           => ( ( ( set_real2 @ L2 )
                = A4 )
             => ( ( size_size_list_real @ L2 )
               != ( finite_card_real @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5340_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ~ ! [L2: list_rat] :
            ( ( sorted_wrt_rat @ ord_less_rat @ L2 )
           => ( ( ( set_rat2 @ L2 )
                = A4 )
             => ( ( size_size_list_rat @ L2 )
               != ( finite_card_rat @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5341_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ~ ! [L2: list_num] :
            ( ( sorted_wrt_num @ ord_less_num @ L2 )
           => ( ( ( set_num2 @ L2 )
                = A4 )
             => ( ( size_size_list_num @ L2 )
               != ( finite_card_num @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5342_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ~ ! [L2: list_nat] :
            ( ( sorted_wrt_nat @ ord_less_nat @ L2 )
           => ( ( ( set_nat2 @ L2 )
                = A4 )
             => ( ( size_size_list_nat @ L2 )
               != ( finite_card_nat @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5343_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ~ ! [L2: list_int] :
            ( ( sorted_wrt_int @ ord_less_int @ L2 )
           => ( ( ( set_int2 @ L2 )
                = A4 )
             => ( ( size_size_list_int @ L2 )
               != ( finite_card_int @ A4 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5344_nth__non__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5345_nth__non__equal__first__eq,axiom,
    ! [X: vEBT_VEBTi,Y: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_VEBT_VEBTi @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5346_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5347_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5348_nth__non__equal__first__eq,axiom,
    ! [X: $o,Y: $o,Xs: list_o,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_o @ ( cons_o @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_o @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5349_nth__equal__first__eq,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] :
      ( ~ ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5350_nth__equal__first__eq,axiom,
    ! [X: set_nat,Xs: list_set_nat,N: nat] :
      ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( ( ( nth_set_nat @ ( cons_set_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5351_nth__equal__first__eq,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5352_nth__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5353_nth__equal__first__eq,axiom,
    ! [X: real,Xs: list_real,N: nat] :
      ( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( ( ( nth_real @ ( cons_real @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5354_nth__equal__first__eq,axiom,
    ! [X: $o,Xs: list_o,N: nat] :
      ( ~ ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( ( ( nth_o @ ( cons_o @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5355_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5356_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( cons_int @ X @ ( slice_int @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( slice_int @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_5357_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( slice_nat @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( slice_nat @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_5358_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: $o,Xs: list_o] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_o @ Begin @ End @ ( cons_o @ X @ Xs ) )
          = ( cons_o @ X @ ( slice_o @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_o @ Begin @ End @ ( cons_o @ X @ Xs ) )
          = ( slice_o @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_5359_length__Cons,axiom,
    ! [X: int,Xs: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
      = ( suc @ ( size_size_list_int @ Xs ) ) ) ).

% length_Cons
thf(fact_5360_length__Cons,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) )
      = ( suc @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_Cons
thf(fact_5361_length__Cons,axiom,
    ! [X: real,Xs: list_real] :
      ( ( size_size_list_real @ ( cons_real @ X @ Xs ) )
      = ( suc @ ( size_size_list_real @ Xs ) ) ) ).

% length_Cons
thf(fact_5362_length__Cons,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X @ Xs ) )
      = ( suc @ ( size_size_list_o @ Xs ) ) ) ).

% length_Cons
thf(fact_5363_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_5364_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5365_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > nat > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5366_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5367_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > int > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5368_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: int > vEBT_VEBT > assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2018189785592951398T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2018189785592951398T_VEBT @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5369_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: int > nat > assn,Xs: list_int,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8891422820522952478nt_nat @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5370_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: int > vEBT_VEBTi > assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L114188773329725699_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L114188773329725699_VEBTi @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5371_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: int > int > assn,Xs: list_int,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8888932350013902202nt_int @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5372_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5373_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: vEBT_VEBT > nat > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5374_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( linord2324613341767563021nteger @ A4 )
          = ( cons_Code_integer @ ( lattic1063845414844153500nteger @ A4 ) @ ( linord2324613341767563021nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ ( lattic1063845414844153500nteger @ A4 ) @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sorted_list_of_set_nonempty
thf(fact_5375_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( linord2612477271533052124et_int @ A4 )
          = ( cons_int @ ( lattic8718645017227715691in_int @ A4 ) @ ( linord2612477271533052124et_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ ( lattic8718645017227715691in_int @ A4 ) @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sorted_list_of_set_nonempty
thf(fact_5376_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( linord3142498349692569832_set_o @ A4 )
          = ( cons_o @ ( lattic1973801136483472281_Min_o @ A4 ) @ ( linord3142498349692569832_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ ( lattic1973801136483472281_Min_o @ A4 ) @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sorted_list_of_set_nonempty
thf(fact_5377_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( linord2614967742042102400et_nat @ A4 )
          = ( cons_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sorted_list_of_set_nonempty
thf(fact_5378_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_literal,L: list_literal] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( ( sorted_wrt_literal @ ord_less_literal @ L )
          & ( ( set_literal2 @ L )
            = A4 )
          & ( ( size_s2501651207091587910iteral @ L )
            = ( finite_card_literal @ A4 ) ) )
        = ( ( linord2913955441264437540iteral @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5379_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_Code_integer,L: list_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ L )
          & ( ( set_Code_integer2 @ L )
            = A4 )
          & ( ( size_s3445333598471063425nteger @ L )
            = ( finite4902975817058060853nteger @ A4 ) ) )
        = ( ( linord2324613341767563021nteger @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5380_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_o,L: list_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( sorted_wrt_o @ ord_less_o @ L )
          & ( ( set_o2 @ L )
            = A4 )
          & ( ( size_size_list_o @ L )
            = ( finite_card_o @ A4 ) ) )
        = ( ( linord3142498349692569832_set_o @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5381_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_real,L: list_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( sorted_wrt_real @ ord_less_real @ L )
          & ( ( set_real2 @ L )
            = A4 )
          & ( ( size_size_list_real @ L )
            = ( finite_card_real @ A4 ) ) )
        = ( ( linord4252657396651189596t_real @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5382_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_rat,L: list_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( ( sorted_wrt_rat @ ord_less_rat @ L )
          & ( ( set_rat2 @ L )
            = A4 )
          & ( ( size_size_list_rat @ L )
            = ( finite_card_rat @ A4 ) ) )
        = ( ( linord1979837681955606664et_rat @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5383_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_num,L: list_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( ( sorted_wrt_num @ ord_less_num @ L )
          & ( ( set_num2 @ L )
            = A4 )
          & ( ( size_size_list_num @ L )
            = ( finite_card_num @ A4 ) ) )
        = ( ( linord8395671565052656842et_num @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5384_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_int,L: list_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( sorted_wrt_int @ ord_less_int @ L )
          & ( ( set_int2 @ L )
            = A4 )
          & ( ( size_size_list_int @ L )
            = ( finite_card_int @ A4 ) ) )
        = ( ( linord2612477271533052124et_int @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5385_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_nat,L: list_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
          & ( ( set_nat2 @ L )
            = A4 )
          & ( ( size_size_list_nat @ L )
            = ( finite_card_nat @ A4 ) ) )
        = ( ( linord2614967742042102400et_nat @ A4 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5386_converse__trancl__induct,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,P2: uint32 > $o] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ! [Y3: uint32] :
            ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y3 @ B ) @ R )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: uint32,Z3: uint32] :
              ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y3 @ Z3 ) @ R )
             => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Z3 @ B ) @ ( transi3114468042090999947uint32 @ R ) )
               => ( ( P2 @ Z3 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_5387_converse__trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: nat,Z3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
               => ( ( P2 @ Z3 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_5388_converse__trancl__induct,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,P2: int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ! [Y3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ B ) @ R )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: int,Z3: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z3 ) @ R )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ B ) @ ( transi6261509568448316235cl_int @ R ) )
               => ( ( P2 @ Z3 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_5389_trancl__trans__induct,axiom,
    ! [X: uint32,Y: uint32,R: set_Pr1773385645901665561uint32,P2: uint32 > uint32 > $o] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ! [X3: uint32,Y3: uint32] :
            ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ Y3 ) @ R )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: uint32,Y3: uint32,Z3: uint32] :
              ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ Y3 ) @ ( transi3114468042090999947uint32 @ R ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y3 @ Z3 ) @ ( transi3114468042090999947uint32 @ R ) )
                 => ( ( P2 @ Y3 @ Z3 )
                   => ( P2 @ X3 @ Z3 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_5390_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,P2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ ( transi6264000038957366511cl_nat @ R ) )
                 => ( ( P2 @ Y3 @ Z3 )
                   => ( P2 @ X3 @ Z3 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_5391_trancl__trans__induct,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,P2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ! [X3: int,Y3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: int,Y3: int,Z3: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( transi6261509568448316235cl_int @ R ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z3 ) @ ( transi6261509568448316235cl_int @ R ) )
                 => ( ( P2 @ Y3 @ Z3 )
                   => ( P2 @ X3 @ Z3 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_5392_groupy,axiom,
    ! [A4: assn,B4: assn,C3: assn,D4: assn,X6: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ A4 @ B4 ) @ ( times_times_assn @ C3 @ D4 ) ) @ X6 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ A4 @ B4 ) @ C3 ) @ D4 ) @ X6 ) ) ).

% groupy
thf(fact_5393_midextr,axiom,
    ! [P2: assn,Q2: assn,Q4: assn,R2: assn,X6: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ P2 @ Q2 ) @ Q4 ) @ R2 ) @ X6 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ P2 @ R2 ) @ Q2 ) @ Q4 ) @ X6 ) ) ).

% midextr
thf(fact_5394_swappa,axiom,
    ! [B4: assn,A4: assn,C3: assn,X6: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ B4 @ A4 ) @ C3 ) @ X6 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ A4 @ B4 ) @ C3 ) @ X6 ) ) ).

% swappa
thf(fact_5395_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( set_real2 @ ( linord4252657396651189596t_real @ A4 ) )
        = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_5396_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( set_o2 @ ( linord3142498349692569832_set_o @ A4 ) )
        = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_5397_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( set_int2 @ ( linord2612477271533052124et_int @ A4 ) )
        = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_5398_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( set_Code_integer2 @ ( linord2324613341767563021nteger @ A4 ) )
        = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_5399_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A4 ) )
        = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_5400_ent__false__iff,axiom,
    ! [P2: assn] :
      ( ( entails @ P2 @ bot_bot_assn )
      = ( ! [H4: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P2 @ H4 ) ) ) ).

% ent_false_iff
thf(fact_5401_ent__trans,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( ( entails @ P2 @ Q2 )
     => ( ( entails @ Q2 @ R2 )
       => ( entails @ P2 @ R2 ) ) ) ).

% ent_trans
thf(fact_5402_ent__refl,axiom,
    ! [P2: assn] : ( entails @ P2 @ P2 ) ).

% ent_refl
thf(fact_5403_ent__iffI,axiom,
    ! [A4: assn,B4: assn] :
      ( ( entails @ A4 @ B4 )
     => ( ( entails @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% ent_iffI
thf(fact_5404_ent__star__mono,axiom,
    ! [P2: assn,P6: assn,Q2: assn,Q4: assn] :
      ( ( entails @ P2 @ P6 )
     => ( ( entails @ Q2 @ Q4 )
       => ( entails @ ( times_times_assn @ P2 @ Q2 ) @ ( times_times_assn @ P6 @ Q4 ) ) ) ) ).

% ent_star_mono
thf(fact_5405_entails__def,axiom,
    ( entails
    = ( ^ [P5: assn,Q6: assn] :
        ! [H4: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P5 @ H4 )
         => ( rep_assn @ Q6 @ H4 ) ) ) ) ).

% entails_def
thf(fact_5406_entailsI,axiom,
    ! [P2: assn,Q2: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H )
         => ( rep_assn @ Q2 @ H ) )
     => ( entails @ P2 @ Q2 ) ) ).

% entailsI
thf(fact_5407_entailsD,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( ( entails @ P2 @ Q2 )
     => ( ( rep_assn @ P2 @ H2 )
       => ( rep_assn @ Q2 @ H2 ) ) ) ).

% entailsD
thf(fact_5408_ent__fwd,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat,Q2: assn] :
      ( ( rep_assn @ P2 @ H2 )
     => ( ( entails @ P2 @ Q2 )
       => ( rep_assn @ Q2 @ H2 ) ) ) ).

% ent_fwd
thf(fact_5409_ent__false,axiom,
    ! [P2: assn] : ( entails @ bot_bot_assn @ P2 ) ).

% ent_false
thf(fact_5410_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( linord2612477271533052124et_int @ A4 )
        = ( linord2612477271533052124et_int @ B4 ) )
     => ( ( finite_finite_int @ A4 )
       => ( ( finite_finite_int @ B4 )
         => ( A4 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_5411_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( ( linord2324613341767563021nteger @ A4 )
        = ( linord2324613341767563021nteger @ B4 ) )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( finite6017078050557962740nteger @ B4 )
         => ( A4 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_5412_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( linord2614967742042102400et_nat @ A4 )
        = ( linord2614967742042102400et_nat @ B4 ) )
     => ( ( finite_finite_nat @ A4 )
       => ( ( finite_finite_nat @ B4 )
         => ( A4 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_5413_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_rat] : ( sorted_wrt_rat @ ord_less_eq_rat @ ( linord1979837681955606664et_rat @ A4 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5414_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_num] : ( sorted_wrt_num @ ord_less_eq_num @ ( linord8395671565052656842et_num @ A4 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5415_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_int] : ( sorted_wrt_int @ ord_less_eq_int @ ( linord2612477271533052124et_int @ A4 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5416_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5417_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_real] : ( sorted_wrt_real @ ord_less_real @ ( linord4252657396651189596t_real @ A4 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5418_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_rat] : ( sorted_wrt_rat @ ord_less_rat @ ( linord1979837681955606664et_rat @ A4 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5419_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_num] : ( sorted_wrt_num @ ord_less_num @ ( linord8395671565052656842et_num @ A4 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5420_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_int] : ( sorted_wrt_int @ ord_less_int @ ( linord2612477271533052124et_int @ A4 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5421_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5422_trancl__induct2,axiom,
    ! [Ax: uint32,Ay: uint32,Bx: uint32,By: uint32,R: set_Pr3773659940955823943uint32,P2: uint32 > uint32 > $o] :
      ( ( member4724532482705224080uint32 @ ( produc6884863695460953815uint32 @ ( produc1400373151660368625uint32 @ Ax @ Ay ) @ ( produc1400373151660368625uint32 @ Bx @ By ) ) @ ( transi5456044692924788698uint32 @ R ) )
     => ( ! [A3: uint32,B3: uint32] :
            ( ( member4724532482705224080uint32 @ ( produc6884863695460953815uint32 @ ( produc1400373151660368625uint32 @ Ax @ Ay ) @ ( produc1400373151660368625uint32 @ A3 @ B3 ) ) @ R )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: uint32,B3: uint32,Aa: uint32,Ba: uint32] :
              ( ( member4724532482705224080uint32 @ ( produc6884863695460953815uint32 @ ( produc1400373151660368625uint32 @ Ax @ Ay ) @ ( produc1400373151660368625uint32 @ A3 @ B3 ) ) @ ( transi5456044692924788698uint32 @ R ) )
             => ( ( member4724532482705224080uint32 @ ( produc6884863695460953815uint32 @ ( produc1400373151660368625uint32 @ A3 @ B3 ) @ ( produc1400373151660368625uint32 @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5423_trancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ R )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ ( transi243908449541399842at_nat @ R ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5424_trancl__induct2,axiom,
    ! [Ax: int,Ay: int,Bx: int,By: int,R: set_Pr2560585780119916871nt_int,P2: int > int > $o] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi6288783178788033498nt_int @ R ) )
     => ( ! [A3: int,B3: int] :
            ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A3 @ B3 ) ) @ R )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: int,B3: int,Aa: int,Ba: int] :
              ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A3 @ B3 ) ) @ ( transi6288783178788033498nt_int @ R ) )
             => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5425_trancl__induct2,axiom,
    ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R: set_Pr7928877670098842301et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi3145040225084697757et_nat @ R ) )
     => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
            ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A3 @ B3 ) ) @ R )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
              ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A3 @ B3 ) ) @ ( transi3145040225084697757et_nat @ R ) )
             => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A3 @ B3 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5426_trancl__induct2,axiom,
    ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R: set_Pr3444600963470892981et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi5221092739591632921et_nat @ R ) )
     => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
            ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A3 @ B3 ) ) @ R )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
              ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A3 @ B3 ) ) @ ( transi5221092739591632921et_nat @ R ) )
             => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A3 @ B3 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5427_trancl_Ocases,axiom,
    ! [A12: uint32,A23: uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A12 @ A23 ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A12 @ A23 ) @ R )
       => ~ ! [B3: uint32] :
              ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A12 @ B3 ) @ ( transi3114468042090999947uint32 @ R ) )
             => ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ B3 @ A23 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_5428_trancl_Ocases,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ R )
       => ~ ! [B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ B3 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A23 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_5429_trancl_Ocases,axiom,
    ! [A12: int,A23: int,R: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A12 @ A23 ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A12 @ A23 ) @ R )
       => ~ ! [B3: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A12 @ B3 ) @ ( transi6261509568448316235cl_int @ R ) )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B3 @ A23 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_5430_trancl_Osimps,axiom,
    ! [A12: uint32,A23: uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A12 @ A23 ) @ ( transi3114468042090999947uint32 @ R ) )
      = ( ? [A7: uint32,B7: uint32] :
            ( ( A12 = A7 )
            & ( A23 = B7 )
            & ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A7 @ B7 ) @ R ) )
        | ? [A7: uint32,B7: uint32,C6: uint32] :
            ( ( A12 = A7 )
            & ( A23 = C6 )
            & ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A7 @ B7 ) @ ( transi3114468042090999947uint32 @ R ) )
            & ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ B7 @ C6 ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_5431_trancl_Osimps,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( transi6264000038957366511cl_nat @ R ) )
      = ( ? [A7: nat,B7: nat] :
            ( ( A12 = A7 )
            & ( A23 = B7 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B7 ) @ R ) )
        | ? [A7: nat,B7: nat,C6: nat] :
            ( ( A12 = A7 )
            & ( A23 = C6 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B7 ) @ ( transi6264000038957366511cl_nat @ R ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B7 @ C6 ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_5432_trancl_Osimps,axiom,
    ! [A12: int,A23: int,R: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A12 @ A23 ) @ ( transi6261509568448316235cl_int @ R ) )
      = ( ? [A7: int,B7: int] :
            ( ( A12 = A7 )
            & ( A23 = B7 )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A7 @ B7 ) @ R ) )
        | ? [A7: int,B7: int,C6: int] :
            ( ( A12 = A7 )
            & ( A23 = C6 )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A7 @ B7 ) @ ( transi6261509568448316235cl_int @ R ) )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B7 @ C6 ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_5433_trancl_Or__into__trancl,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ R )
     => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ ( transi3114468042090999947uint32 @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_5434_trancl_Or__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_5435_trancl_Or__into__trancl,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_5436_tranclE,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ R )
       => ~ ! [C: uint32] :
              ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ C ) @ ( transi3114468042090999947uint32 @ R ) )
             => ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ C @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_5437_tranclE,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ~ ! [C: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_5438_tranclE,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
       => ~ ! [C: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C ) @ ( transi6261509568448316235cl_int @ R ) )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ C @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_5439_trancl__trans,axiom,
    ! [X: uint32,Y: uint32,R: set_Pr1773385645901665561uint32,Z: uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y @ Z ) @ ( transi3114468042090999947uint32 @ R ) )
       => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Z ) @ ( transi3114468042090999947uint32 @ R ) ) ) ) ).

% trancl_trans
thf(fact_5440_trancl__trans,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% trancl_trans
thf(fact_5441_trancl__trans,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Z: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi6261509568448316235cl_int @ R ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R ) ) ) ) ).

% trancl_trans
thf(fact_5442_trancl__induct,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,P2: uint32 > $o] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ! [Y3: uint32] :
            ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ Y3 ) @ R )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: uint32,Z3: uint32] :
              ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ Y3 ) @ ( transi3114468042090999947uint32 @ R ) )
             => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y3 @ Z3 ) @ R )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z3 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_5443_trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: nat,Z3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z3 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_5444_trancl__induct,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,P2: int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ! [Y3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ Y3 ) @ R )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: int,Z3: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ Y3 ) @ ( transi6261509568448316235cl_int @ R ) )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z3 ) @ R )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z3 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_5445_r__r__into__trancl,axiom,
    ! [A: uint32,B: uint32,R2: set_Pr1773385645901665561uint32,C2: uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ R2 )
     => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ B @ C2 ) @ R2 )
       => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ C2 ) @ ( transi3114468042090999947uint32 @ R2 ) ) ) ) ).

% r_r_into_trancl
thf(fact_5446_r__r__into__trancl,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% r_r_into_trancl
thf(fact_5447_r__r__into__trancl,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,C2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C2 ) @ R2 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C2 ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).

% r_r_into_trancl
thf(fact_5448_converse__tranclE,axiom,
    ! [X: uint32,Z: uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Z ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Z ) @ R )
       => ~ ! [Y3: uint32] :
              ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y3 ) @ R )
             => ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y3 @ Z ) @ ( transi3114468042090999947uint32 @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_5449_converse__tranclE,axiom,
    ! [X: nat,Z: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R )
       => ~ ! [Y3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_5450_converse__tranclE,axiom,
    ! [X: int,Z: int,R: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R )
       => ~ ! [Y3: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z ) @ ( transi6261509568448316235cl_int @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_5451_irrefl__trancl__rD,axiom,
    ! [R: set_Pr1773385645901665561uint32,X: uint32,Y: uint32] :
      ( ! [X3: uint32] :
          ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ X3 ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_5452_irrefl__trancl__rD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_5453_irrefl__trancl__rD,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_5454_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,C2: uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ ( transi3114468042090999947uint32 @ R ) )
     => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ B @ C2 ) @ R )
       => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ C2 ) @ ( transi3114468042090999947uint32 @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_5455_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_5456_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,C2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C2 ) @ R )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C2 ) @ ( transi6261509568448316235cl_int @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_5457_trancl__into__trancl2,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,C2: uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ R )
     => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ B @ C2 ) @ ( transi3114468042090999947uint32 @ R ) )
       => ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ C2 ) @ ( transi3114468042090999947uint32 @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_5458_trancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( transi6264000038957366511cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_5459_trancl__into__trancl2,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,C2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C2 ) @ ( transi6261509568448316235cl_int @ R ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C2 ) @ ( transi6261509568448316235cl_int @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_5460_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( 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 @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) ) ) ) ).

% local.ext
thf(fact_5461_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: produc3658429121746597890et_nat,P2: assn,R2: assn,F2: assn] :
      ( ( rep_assn @ Ps @ H2 )
     => ( ( entails @ P2 @ R2 )
       => ( ( entails @ Ps @ ( times_times_assn @ P2 @ F2 ) )
         => ( rep_assn @ ( times_times_assn @ R2 @ F2 ) @ H2 ) ) ) ) ).

% mod_frame_fwd
thf(fact_5462_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: 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_is ) @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) ) @ ( 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 ) ) ) ).

% assnle
thf(fact_5463_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5464_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > nat > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5465_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5466_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > int > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5467_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: int > vEBT_VEBT > assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2018189785592951398T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L2018189785592951398T_VEBT @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5468_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: int > nat > assn,Xs: list_int,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L8891422820522952478nt_nat @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5469_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: int > vEBT_VEBTi > assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L114188773329725699_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L114188773329725699_VEBTi @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5470_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: int > int > assn,Xs: list_int,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L8888932350013902202nt_int @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5471_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5472_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: vEBT_VEBT > nat > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A4 @ Xs @ Xsi ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5473_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBTi > assn,X: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5474_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBT > assn,X: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5475_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: vEBT_VEBT > vEBT_VEBT > assn,X: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5476_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: real > vEBT_VEBTi > assn,X: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A4 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5477_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: real > vEBT_VEBT > assn,X: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A4 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5478_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: $o > vEBT_VEBTi > assn,X: $o,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A4 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5479_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: $o > vEBT_VEBT > assn,X: $o,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A4 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5480_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: nat > vEBT_VEBTi > assn,X: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A4 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5481_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: nat > vEBT_VEBT > assn,X: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A4 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5482_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: vEBT_VEBT > vEBT_VEBTi > assn,X: vEBT_VEBT,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ A4 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ Q2 )
           => ( entails @ P2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5483_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: 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_is @ 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_is ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( 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 ) ) ) ) ).

% recomp
thf(fact_5484_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: 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_is @ I ) ) @ Rest ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( 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_is ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% repack
thf(fact_5485_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_5486_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_set_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5487_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_real @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5488_atLeastLessThan__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I @ ( set_or8585797421378605585et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I )
        & ( ord_less_set_int @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5489_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_rat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5490_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_num @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5491_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5492_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_int @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5493_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or8404916559141939852nteger @ L @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L @ I )
        & ( ord_le6747313008572928689nteger @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_5494_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_eq_o @ B @ A )
     => ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
thf(fact_5495_atLeastLessThan__empty,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( set_or8585797421378605585et_int @ A @ B )
        = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_5496_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( set_or4029947393144176647an_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
thf(fact_5497_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
thf(fact_5498_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_5499_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_5500_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_5501_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or7139685690850216873Than_o @ A @ B ) )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5502_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or66887138388493659n_real @ A @ B ) )
      = ( ~ ( ord_less_real @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5503_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or4029947393144176647an_rat @ A @ B ) )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5504_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or1222409239386451017an_num @ A @ B ) )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5505_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5506_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or4662586982721622107an_int @ A @ B ) )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5507_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or8404916559141939852nteger @ A @ B ) )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5508_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5509_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_real @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5510_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5511_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5512_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5513_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5514_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5515_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Ico_iff
thf(fact_5516_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Ico_iff
thf(fact_5517_ivl__subset,axiom,
    ! [I: rat,J: rat,M: rat,N: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I @ J ) @ ( set_or4029947393144176647an_rat @ M @ N ) )
      = ( ( ord_less_eq_rat @ J @ I )
        | ( ( ord_less_eq_rat @ M @ I )
          & ( ord_less_eq_rat @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_5518_ivl__subset,axiom,
    ! [I: num,J: num,M: num,N: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J ) @ ( set_or1222409239386451017an_num @ M @ N ) )
      = ( ( ord_less_eq_num @ J @ I )
        | ( ( ord_less_eq_num @ M @ I )
          & ( ord_less_eq_num @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_5519_ivl__subset,axiom,
    ! [I: nat,J: nat,M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ J @ I )
        | ( ( ord_less_eq_nat @ M @ I )
          & ( ord_less_eq_nat @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_5520_ivl__subset,axiom,
    ! [I: int,J: int,M: int,N: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
      = ( ( ord_less_eq_int @ J @ I )
        | ( ( ord_less_eq_int @ M @ I )
          & ( ord_less_eq_int @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_5521_ivl__subset,axiom,
    ! [I: code_integer,J: code_integer,M: code_integer,N: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ I @ J ) @ ( set_or8404916559141939852nteger @ M @ N ) )
      = ( ( ord_le3102999989581377725nteger @ J @ I )
        | ( ( ord_le3102999989581377725nteger @ M @ I )
          & ( ord_le3102999989581377725nteger @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_5522_ivl__diff,axiom,
    ! [I: rat,N: rat,M: rat] :
      ( ( ord_less_eq_rat @ I @ N )
     => ( ( minus_minus_set_rat @ ( set_or4029947393144176647an_rat @ I @ M ) @ ( set_or4029947393144176647an_rat @ I @ N ) )
        = ( set_or4029947393144176647an_rat @ N @ M ) ) ) ).

% ivl_diff
thf(fact_5523_ivl__diff,axiom,
    ! [I: num,N: num,M: num] :
      ( ( ord_less_eq_num @ I @ N )
     => ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N ) )
        = ( set_or1222409239386451017an_num @ N @ M ) ) ) ).

% ivl_diff
thf(fact_5524_ivl__diff,axiom,
    ! [I: nat,N: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
        = ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).

% ivl_diff
thf(fact_5525_ivl__diff,axiom,
    ! [I: int,N: int,M: int] :
      ( ( ord_less_eq_int @ I @ N )
     => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
        = ( set_or4662586982721622107an_int @ N @ M ) ) ) ).

% ivl_diff
thf(fact_5526_ivl__diff,axiom,
    ! [I: code_integer,N: code_integer,M: code_integer] :
      ( ( ord_le3102999989581377725nteger @ I @ N )
     => ( ( minus_2355218937544613996nteger @ ( set_or8404916559141939852nteger @ I @ M ) @ ( set_or8404916559141939852nteger @ I @ N ) )
        = ( set_or8404916559141939852nteger @ N @ M ) ) ) ).

% ivl_diff
thf(fact_5527_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ I )
     => ( ( list_u6098035379799741383_VEBTi @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_5528_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_5529_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,X: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ I )
     => ( ( list_update_real @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_5530_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
     => ( ( list_update_o @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_5531_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_5532_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_5533_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_VEBT_VEBT,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( 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 @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5534_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_real,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_real @ TreeList3 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( 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 @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5535_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_o,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_o @ TreeList3 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( 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 @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5536_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_nat,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_nat @ TreeList3 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( 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 @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5537_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,X: int] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( nth_int @ ( list_update_int @ L @ J @ X ) @ I )
        = ( nth_int @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_5538_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,X: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ J @ X ) @ I )
        = ( nth_VEBT_VEBTi @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_5539_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ~ ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ J @ X ) @ I )
        = ( nth_VEBT_VEBT @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_5540_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,X: real] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( nth_real @ ( list_update_real @ L @ J @ X ) @ I )
        = ( nth_real @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_5541_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J: nat,X: $o] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( nth_o @ ( list_update_o @ L @ J @ X ) @ I )
        = ( nth_o @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_5542_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,X: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( list_update_nat @ L @ J @ X ) @ I )
        = ( nth_nat @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_5543_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_5544_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_5545_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_5546_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( nth_real @ ( list_update_real @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_5547_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_5548_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_5549_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_5550_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( 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 @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( 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 ) ) ) ) ).

% txe
thf(fact_5551_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( set_int2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_5552_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ ( nth_VEBT_VEBTi @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBTi @ Xs @ I ) ) )
          = ( set_VEBT_VEBTi2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_5553_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_5554_set__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
       => ( ( set_real2 @ ( list_update_real @ ( list_update_real @ Xs @ I @ ( nth_real @ Xs @ J ) ) @ J @ ( nth_real @ Xs @ I ) ) )
          = ( set_real2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_5555_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
          = ( set_o2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_5556_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_5557_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > uint32] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups833757482993574392uint32 @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_uint32 ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups833757482993574392uint32 @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_uint32 @ ( groups833757482993574392uint32 @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_5558_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_5559_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_5560_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_5561_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_5562_nth__image__indices,axiom,
    ! [L: list_VEBT_VEBTi] :
      ( ( image_nat_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ L ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ L ) ) )
      = ( set_VEBT_VEBTi2 @ L ) ) ).

% nth_image_indices
thf(fact_5563_nth__image__indices,axiom,
    ! [L: list_set_nat] :
      ( ( image_nat_set_nat @ ( nth_set_nat @ L ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ L ) ) )
      = ( set_set_nat2 @ L ) ) ).

% nth_image_indices
thf(fact_5564_nth__image__indices,axiom,
    ! [L: list_int] :
      ( ( image_nat_int @ ( nth_int @ L ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ L ) ) )
      = ( set_int2 @ L ) ) ).

% nth_image_indices
thf(fact_5565_nth__image__indices,axiom,
    ! [L: list_VEBT_VEBT] :
      ( ( image_nat_VEBT_VEBT @ ( nth_VEBT_VEBT @ L ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ L ) ) )
      = ( set_VEBT_VEBT2 @ L ) ) ).

% nth_image_indices
thf(fact_5566_nth__image__indices,axiom,
    ! [L: list_real] :
      ( ( image_nat_real @ ( nth_real @ L ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ L ) ) )
      = ( set_real2 @ L ) ) ).

% nth_image_indices
thf(fact_5567_nth__image__indices,axiom,
    ! [L: list_o] :
      ( ( image_nat_o @ ( nth_o @ L ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ L ) ) )
      = ( set_o2 @ L ) ) ).

% nth_image_indices
thf(fact_5568_nth__image__indices,axiom,
    ! [L: list_nat] :
      ( ( image_nat_nat @ ( nth_nat @ L ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ L ) ) )
      = ( set_nat2 @ L ) ) ).

% nth_image_indices
thf(fact_5569_deleti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V1365221501068881998eletei @ T @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ T @ X ) ) ) ) ).

% deleti'_rf_abstr
thf(fact_5570_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ( ( set_or66887138388493659n_real @ A @ B )
            = ( set_or66887138388493659n_real @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_5571_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ( ( set_or4029947393144176647an_rat @ A @ B )
            = ( set_or4029947393144176647an_rat @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_5572_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ C2 @ D )
       => ( ( ( set_or1222409239386451017an_num @ A @ B )
            = ( set_or1222409239386451017an_num @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_5573_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_5574_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_5575_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D )
       => ( ( ( set_or8404916559141939852nteger @ A @ B )
            = ( set_or8404916559141939852nteger @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_5576_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C2 @ D ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_5577_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C2 @ D ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_5578_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C2 @ D ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_5579_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C2 @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_5580_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C2 @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_5581_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C2 @ D ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_5582_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C2 @ D ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_5583_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C2 @ D ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_5584_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C2 @ D ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_5585_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C2 @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_5586_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C2 @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_5587_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C2 @ D ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_5588_listI__assn__conv,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( N
        = ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) @ A4 @ Xs @ Xsi )
        = ( vEBT_L6296928887356842470_VEBTi @ A4 @ Xs @ Xsi ) ) ) ).

% listI_assn_conv
thf(fact_5589_list__assn__conv__idx,axiom,
    ( vEBT_L6296928887356842470_VEBTi
    = ( ^ [A6: vEBT_VEBT > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBT] : ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ A6 @ Xs2 ) ) ) ).

% list_assn_conv_idx
thf(fact_5590_list__update__code_I3_J,axiom,
    ! [X: int,Xs: list_int,I: nat,Y: int] :
      ( ( list_update_int @ ( cons_int @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_int @ X @ ( list_update_int @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_5591_list__update__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,I: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_5592_list__update__code_I3_J,axiom,
    ! [X: $o,Xs: list_o,I: nat,Y: $o] :
      ( ( list_update_o @ ( cons_o @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_o @ X @ ( list_update_o @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_5593_list__update__code_I3_J,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_VEBT_VEBTi @ X @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_5594_list__update__code_I3_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_VEBT_VEBT @ X @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_5595_list__update__code_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ( list_update_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_int @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_5596_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_5597_list__update__code_I2_J,axiom,
    ! [X: $o,Xs: list_o,Y: $o] :
      ( ( list_update_o @ ( cons_o @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_o @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_5598_list__update__code_I2_J,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( cons_VEBT_VEBTi @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_VEBT_VEBTi @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_5599_list__update__code_I2_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_VEBT_VEBT @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_5600_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A4: set_set_nat,X: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_5601_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_5602_set__update__subsetI,axiom,
    ! [Xs: list_real,A4: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_5603_set__update__subsetI,axiom,
    ! [Xs: list_o,A4: set_o,X: $o,I: nat] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_5604_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A4: set_VEBT_VEBTi,X: vEBT_VEBTi,I: nat] :
      ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs ) @ A4 )
     => ( ( member_VEBT_VEBTi @ X @ A4 )
       => ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_5605_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_5606_set__update__subsetI,axiom,
    ! [Xs: list_int,A4: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_5607_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_5608_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_5609_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C2 @ D ) )
     => ( ( ord_less_eq_rat @ B @ A )
        | ( ( ord_less_eq_rat @ C2 @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_5610_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C2 @ D ) )
     => ( ( ord_less_eq_num @ B @ A )
        | ( ( ord_less_eq_num @ C2 @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_5611_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C2 @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_5612_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C2 @ D ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C2 @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_5613_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C2 @ D ) )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
        | ( ( ord_le3102999989581377725nteger @ C2 @ A )
          & ( ord_le3102999989581377725nteger @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_5614_ex__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
            & ( P2 @ M5 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_5615_all__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
           => ( P2 @ M5 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_5616_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_5617_listI__assn__conv_H,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi,F2: assn] :
      ( ( N
        = ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) @ A4 @ Xs @ Xsi ) @ F2 )
        = ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ A4 @ Xs @ Xsi ) @ F2 ) ) ) ).

% listI_assn_conv'
thf(fact_5618_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N8: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N8 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N8 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_5619_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) )
     => ( A4
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_5620_extract__pre__list__assn__lengthD,axiom,
    ! [A4: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1279224858307276611T_VEBT @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5621_extract__pre__list__assn__lengthD,axiom,
    ! [A4: real > vEBT_VEBT > assn,Xs: list_real,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4595930785310033027T_VEBT @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_size_list_real @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5622_extract__pre__list__assn__lengthD,axiom,
    ! [A4: $o > vEBT_VEBT > assn,Xs: list_o,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1750719106661372127T_VEBT @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_size_list_o @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5623_extract__pre__list__assn__lengthD,axiom,
    ! [A4: nat > vEBT_VEBT > assn,Xs: list_nat,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L8158188754432654943T_VEBT @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_size_list_nat @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5624_extract__pre__list__assn__lengthD,axiom,
    ! [A4: vEBT_VEBT > real > assn,Xs: list_VEBT_VEBT,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L5781919052683127133T_real @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5625_extract__pre__list__assn__lengthD,axiom,
    ! [A4: real > real > assn,Xs: list_real,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1930518968523514909l_real @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_real @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5626_extract__pre__list__assn__lengthD,axiom,
    ! [A4: $o > real > assn,Xs: list_o,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4725278957065240257o_real @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_o @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5627_extract__pre__list__assn__lengthD,axiom,
    ! [A4: nat > real > assn,Xs: list_nat,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6102073776069194049t_real @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_nat @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5628_extract__pre__list__assn__lengthD,axiom,
    ! [A4: vEBT_VEBT > $o > assn,Xs: list_VEBT_VEBT,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L7489408758114837031VEBT_o @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5629_extract__pre__list__assn__lengthD,axiom,
    ! [A4: real > $o > assn,Xs: list_real,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6234343332106409831real_o @ A4 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_real @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5630_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_5631_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_5632_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_5633_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_5634_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_5635_set__update__memI,axiom,
    ! [N: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_5636_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_5637_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_int @ X @ ( set_int2 @ L ) )
            & ! [Y4: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_5638_in__set__upd__eq,axiom,
    ! [I: nat,L: list_set_nat,X: set_nat,Y: set_nat] :
      ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ L ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_set_nat @ X @ ( set_set_nat2 @ L ) )
            & ! [Y4: set_nat] : ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_5639_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) )
            & ! [Y4: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_5640_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) )
            & ! [Y4: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_5641_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_real @ X @ ( set_real2 @ L ) )
            & ! [Y4: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_5642_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_o @ X @ ( set_o2 @ L ) )
            & ! [Y4: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_5643_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_nat @ X @ ( set_nat2 @ L ) )
            & ! [Y4: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_5644_in__set__upd__cases,axiom,
    ! [X: int,L: list_int,I: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
         => ( X != Y ) )
       => ( member_int @ X @ ( set_int2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_5645_in__set__upd__cases,axiom,
    ! [X: set_nat,L: list_set_nat,I: nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ L ) )
         => ( X != Y ) )
       => ( member_set_nat @ X @ ( set_set_nat2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_5646_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_5647_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_5648_in__set__upd__cases,axiom,
    ! [X: real,L: list_real,I: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
         => ( X != Y ) )
       => ( member_real @ X @ ( set_real2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_5649_in__set__upd__cases,axiom,
    ! [X: $o,L: list_o,I: nat,Y: $o] :
      ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
         => ( X = ~ Y ) )
       => ( member_o @ X @ ( set_o2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_5650_in__set__upd__cases,axiom,
    ! [X: nat,L: list_nat,I: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
         => ( X != Y ) )
       => ( member_nat @ X @ ( set_nat2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_5651_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y4: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_5652_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_set_nat,X: set_nat,Y: set_nat] :
      ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ L ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y4: set_nat] : ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_5653_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y4: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_5654_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y4: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_5655_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y4: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_5656_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y4: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_5657_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y4: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_5658_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_int,X: int] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X ) @ J )
          = ( nth_int @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_5659_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X ) @ J )
          = ( nth_VEBT_VEBTi @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_5660_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) @ J )
          = ( nth_VEBT_VEBT @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_5661_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_real,X: real] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X ) @ J )
          = ( nth_real @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_5662_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,X: $o,J: nat] :
      ( ( nth_o @ ( list_update_o @ L @ I @ X ) @ J )
      = ( ( ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => X )
        & ( ~ ( ( I = J )
              & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => ( nth_o @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_5663_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_nat,X: nat] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X ) @ J )
          = ( nth_nat @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_5664_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_5665_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ( list_u6098035379799741383_VEBTi @ Xs @ I @ X )
          = Xs )
        = ( ( nth_VEBT_VEBTi @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_5666_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
          = Xs )
        = ( ( nth_VEBT_VEBT @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_5667_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( ( list_update_real @ Xs @ I @ X )
          = Xs )
        = ( ( nth_real @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_5668_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ( list_update_o @ Xs @ I @ X )
          = Xs )
        = ( ( nth_o @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_5669_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_5670_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = ( nth_int @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_5671_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ J )
            = ( nth_VEBT_VEBTi @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_5672_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_5673_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,J: nat,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_real @ ( list_update_real @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_real @ ( list_update_real @ Xs @ I @ X ) @ J )
            = ( nth_real @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_5674_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,X: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ J )
        = ( ( ( I = J )
           => X )
          & ( ( I != J )
           => ( nth_o @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_5675_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_5676_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5677_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5678_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,X: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X ) ) @ ( insert_o @ X @ ( set_o2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5679_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] : ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5680_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5681_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5682_sum_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat,G: nat > nat,H2: nat > nat] :
      ( ( A = C2 )
     => ( ( B = D )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C2 @ X3 )
             => ( ( ord_less_nat @ X3 @ D )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups3542108847815614940at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C2 @ D ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_5683_sum_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat,G: nat > real,H2: nat > real] :
      ( ( A = C2 )
     => ( ( B = D )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C2 @ X3 )
             => ( ( ord_less_nat @ X3 @ D )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) ) )
         => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups6591440286371151544t_real @ H2 @ ( set_or4665077453230672383an_nat @ C2 @ D ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_5684_sum_Oivl__cong,axiom,
    ! [A: int,C2: int,B: int,D: int,G: int > int,H2: int > int] :
      ( ( A = C2 )
     => ( ( B = D )
       => ( ! [X3: int] :
              ( ( ord_less_eq_int @ C2 @ X3 )
             => ( ( ord_less_int @ X3 @ D )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) ) )
         => ( ( groups4538972089207619220nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups4538972089207619220nt_int @ H2 @ ( set_or4662586982721622107an_int @ C2 @ D ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_5685_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P3 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) )
          = ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_5686_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P3 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) )
          = ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_5687_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P3 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) )
          = ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_5688_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P3 )
       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) )
          = ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_5689_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P3: nat,F: nat > uint32] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P3 )
       => ( ( minus_minus_uint32 @ ( groups833757482993574392uint32 @ F @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) @ ( groups833757482993574392uint32 @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
          = ( groups833757482993574392uint32 @ F @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_5690_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P3: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P3 )
       => ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
          = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_5691_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P3: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P3 )
       => ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
          = ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_5692_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBTi > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5693_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBT > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5694_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBT > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5695_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: real > vEBT_VEBTi > assn,X1: real,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ I5 @ A4 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L7851252805511451907_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5696_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: real > vEBT_VEBT > assn,X1: real,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ I5 @ A4 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L3095048238742455910T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5697_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: $o > vEBT_VEBTi > assn,X1: $o,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ I5 @ A4 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L6286945158656146733_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5698_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: $o > vEBT_VEBT > assn,X1: $o,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ I5 @ A4 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L1319876754960170684T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5699_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: nat > vEBT_VEBTi > assn,X1: nat,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ I5 @ A4 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L7489483478785760935_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5700_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: nat > vEBT_VEBT > assn,X1: nat,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ I5 @ A4 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L8511957252848910786T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5701_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBTi > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L1528199826722428489_VEBTi @ I5 @ A4 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L1528199826722428489_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5702_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_5703_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N8: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N8 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N8 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_5704_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBTi > assn,X: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5705_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBT > assn,X: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5706_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: vEBT_VEBT > vEBT_VEBT > assn,X: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5707_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: real > vEBT_VEBTi > assn,X: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A4 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5708_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: real > vEBT_VEBT > assn,X: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A4 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5709_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: $o > vEBT_VEBTi > assn,X: $o,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A4 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5710_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: $o > vEBT_VEBT > assn,X: $o,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A4 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5711_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: nat > vEBT_VEBTi > assn,X: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A4 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5712_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: nat > vEBT_VEBT > assn,X: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A4 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5713_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBTi > assn,X: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_Time_Heap_o,Q2: $o > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( A4 @ X @ Xi ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_hoare_triple_o @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5714_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > uint32,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_uint32 )
     => ( ( groups833757482993574392uint32 @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups833757482993574392uint32 @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_5715_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_5716_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_5717_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_5718_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_5719_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_5720_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_5721_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_5722_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_5723_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_5724_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_5725_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_5726_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_5727_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_5728_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_5729_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_5730_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_5731_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_5732_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_5733_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( insert_int @ ( nth_int @ L @ I ) @ ( set_int2 @ ( list_update_int @ L @ I @ X ) ) )
        = ( insert_int @ X @ ( set_int2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5734_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: 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 @ X ) ) )
        = ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5735_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: 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 @ X ) ) )
        = ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5736_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( insert_real @ ( nth_real @ L @ I ) @ ( set_real2 @ ( list_update_real @ L @ I @ X ) ) )
        = ( insert_real @ X @ ( set_real2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5737_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( insert_o @ ( nth_o @ L @ I ) @ ( set_o2 @ ( list_update_o @ L @ I @ X ) ) )
        = ( insert_o @ X @ ( set_o2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5738_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( insert_nat @ ( nth_nat @ L @ I ) @ ( set_nat2 @ ( list_update_nat @ L @ I @ X ) ) )
        = ( insert_nat @ X @ ( set_nat2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5739_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBTi > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5740_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: vEBT_VEBTi > vEBT_VEBT > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5741_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBT > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5742_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: real > vEBT_VEBTi > assn,X1: real,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5743_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: real > vEBT_VEBT > assn,X1: real,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5744_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: $o > vEBT_VEBTi > assn,X1: $o,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5745_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: $o > vEBT_VEBT > assn,X1: $o,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5746_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: nat > vEBT_VEBTi > assn,X1: nat,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5747_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: nat > vEBT_VEBT > assn,X1: nat,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5748_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: vEBT_VEBT > vEBT_VEBTi > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L1528199826722428489_VEBTi @ ( insert_nat @ I @ I5 ) @ A4 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A4 @ X1 @ X2 ) @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ A4 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5749_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is: 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 @ X )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( 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_is ) ) )
          @ ( times_times_assn
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( 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_is @ H2 @ X ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_5750_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,L: nat,X: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: 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 ) @ X )
            @ ( 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_is @ H2 @ X ) ) @ ( 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_is ) ) )
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( 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_is @ H2 @ X ) ) ) ) ) ).

% big_assn_simp
thf(fact_5751_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu2 @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( vEBT_vebt_insert @ Uu2 @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_5752_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_Time_Heap_o,Q2: $o > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu2 @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( vEBT_vebt_insert @ Uu2 @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_hoare_triple_o @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_5753_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T2636463487746394924on_nat,Q2: option_nat > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu2 @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_7629718768684598413on_nat @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( vEBT_vebt_insert @ Uu2 @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_7629718768684598413on_nat @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_5754_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_Time_Heap_nat,Q2: nat > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu2 @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_3067605981109127869le_nat @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( vEBT_vebt_insert @ Uu2 @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F2 ) @ C2 @ Q2 )
           => ( hoare_3067605981109127869le_nat @ P2 @ C2 @ Q2 ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_5755_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBTi,N: nat,X: vEBT_VEBTi] :
      ( ( distinct_VEBT_VEBTi @ Xs )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs @ N @ X ) )
          = ( insert_VEBT_VEBTi @ X @ ( minus_3697805406911847364_VEBTi @ ( set_VEBT_VEBTi2 @ Xs ) @ ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ N ) @ bot_bo8982466882572371071_VEBTi ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_5756_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) )
          = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_5757_set__update__distinct,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( distinct_real @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( ( set_real2 @ ( list_update_real @ Xs @ N @ X ) )
          = ( insert_real @ X @ ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ ( nth_real @ Xs @ N ) @ bot_bot_set_real ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_5758_set__update__distinct,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ Xs @ N @ X ) )
          = ( insert_int @ X @ ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ ( nth_int @ Xs @ N ) @ bot_bot_set_int ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_5759_set__update__distinct,axiom,
    ! [Xs: list_o,N: nat,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ Xs @ N @ X ) )
          = ( insert_o @ X @ ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ ( nth_o @ Xs @ N ) @ bot_bot_set_o ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_5760_set__update__distinct,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_5761_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H )
         => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) )
     => ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q2 ) ) ).

% hoare_triple_preI
thf(fact_5762_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_o,Q2: $o > assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H )
         => ( hoare_hoare_triple_o @ P2 @ C2 @ Q2 ) )
     => ( hoare_hoare_triple_o @ P2 @ C2 @ Q2 ) ) ).

% hoare_triple_preI
thf(fact_5763_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_T2636463487746394924on_nat,Q2: option_nat > assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H )
         => ( hoare_7629718768684598413on_nat @ P2 @ C2 @ Q2 ) )
     => ( hoare_7629718768684598413on_nat @ P2 @ C2 @ Q2 ) ) ).

% hoare_triple_preI
thf(fact_5764_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_nat,Q2: nat > assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H )
         => ( hoare_3067605981109127869le_nat @ P2 @ C2 @ Q2 ) )
     => ( hoare_3067605981109127869le_nat @ P2 @ C2 @ Q2 ) ) ).

% hoare_triple_preI
thf(fact_5765_nth__image,axiom,
    ! [L: nat,Xs: list_VEBT_VEBTi] :
      ( ( ord_less_eq_nat @ L @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( image_nat_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_VEBT_VEBTi2 @ ( take_VEBT_VEBTi @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_5766_nth__image,axiom,
    ! [L: nat,Xs: list_set_nat] :
      ( ( ord_less_eq_nat @ L @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( ( image_nat_set_nat @ ( nth_set_nat @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_set_nat2 @ ( take_set_nat @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_5767_nth__image,axiom,
    ! [L: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_int @ Xs ) )
     => ( ( image_nat_int @ ( nth_int @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_int2 @ ( take_int @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_5768_nth__image,axiom,
    ! [L: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ L @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( image_nat_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_5769_nth__image,axiom,
    ! [L: nat,Xs: list_real] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_real @ Xs ) )
     => ( ( image_nat_real @ ( nth_real @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_real2 @ ( take_real @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_5770_nth__image,axiom,
    ! [L: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_o @ Xs ) )
     => ( ( image_nat_o @ ( nth_o @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_o2 @ ( take_o @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_5771_nth__image,axiom,
    ! [L: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_nat @ Xs ) )
     => ( ( image_nat_nat @ ( nth_nat @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_nat2 @ ( take_nat @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_5772_pure__assn__eq__conv,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ( pure_assn @ P2 )
        = ( pure_assn @ Q2 ) )
      = ( P2 = Q2 ) ) ).

% pure_assn_eq_conv
thf(fact_5773_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_5774_take__update,axiom,
    ! [N: nat,L: list_nat,I: nat,X: nat] :
      ( ( take_nat @ N @ ( list_update_nat @ L @ I @ X ) )
      = ( list_update_nat @ ( take_nat @ N @ L ) @ I @ X ) ) ).

% take_update
thf(fact_5775_take__update,axiom,
    ! [N: nat,L: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( take_VEBT_VEBTi @ N @ ( list_u6098035379799741383_VEBTi @ L @ I @ X ) )
      = ( list_u6098035379799741383_VEBTi @ ( take_VEBT_VEBTi @ N @ L ) @ I @ X ) ) ).

% take_update
thf(fact_5776_take__update,axiom,
    ! [N: nat,L: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( take_VEBT_VEBT @ N @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) )
      = ( list_u1324408373059187874T_VEBT @ ( take_VEBT_VEBT @ N @ L ) @ I @ X ) ) ).

% take_update
thf(fact_5777_pure__assn__eq__emp__iff,axiom,
    ! [P2: $o] :
      ( ( ( pure_assn @ P2 )
        = one_one_assn )
      = P2 ) ).

% pure_assn_eq_emp_iff
thf(fact_5778_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_5779_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_5780_pure__assn__eq__false__iff,axiom,
    ! [P2: $o] :
      ( ( ( pure_assn @ P2 )
        = bot_bot_assn )
      = ~ P2 ) ).

% pure_assn_eq_false_iff
thf(fact_5781_take__Suc__Cons,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( take_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
      = ( cons_int @ X @ ( take_int @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_5782_take__Suc__Cons,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
      = ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_5783_take__Suc__Cons,axiom,
    ! [N: nat,X: $o,Xs: list_o] :
      ( ( take_o @ ( suc @ N ) @ ( cons_o @ X @ Xs ) )
      = ( cons_o @ X @ ( take_o @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_5784_take__all,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N )
     => ( ( take_VEBT_VEBT @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_5785_take__all,axiom,
    ! [Xs: list_real,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ N )
     => ( ( take_real @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_5786_take__all,axiom,
    ! [Xs: list_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N )
     => ( ( take_o @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_5787_take__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( take_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_5788_take__all__iff,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ( take_VEBT_VEBT @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_5789_take__all__iff,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ( take_real @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_5790_take__all__iff,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ( take_o @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_5791_take__all__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( take_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_5792_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBT @ ( take_VEBT_VEBT @ N @ Xs ) @ I )
        = ( nth_VEBT_VEBT @ Xs @ I ) ) ) ).

% nth_take
thf(fact_5793_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBTi @ ( take_VEBT_VEBTi @ N @ Xs ) @ I )
        = ( nth_VEBT_VEBTi @ Xs @ I ) ) ) ).

% nth_take
thf(fact_5794_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_int @ ( take_int @ N @ Xs ) @ I )
        = ( nth_int @ Xs @ I ) ) ) ).

% nth_take
thf(fact_5795_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
        = ( nth_nat @ Xs @ I ) ) ) ).

% nth_take
thf(fact_5796_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_nat,Y: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_nat @ N @ ( list_update_nat @ Xs @ M @ Y ) )
        = ( take_nat @ N @ Xs ) ) ) ).

% take_update_cancel
thf(fact_5797_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_VEBT_VEBTi @ N @ ( list_u6098035379799741383_VEBTi @ Xs @ M @ Y ) )
        = ( take_VEBT_VEBTi @ N @ Xs ) ) ) ).

% take_update_cancel
thf(fact_5798_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_VEBT_VEBT @ N @ ( list_u1324408373059187874T_VEBT @ Xs @ M @ Y ) )
        = ( take_VEBT_VEBT @ N @ Xs ) ) ) ).

% take_update_cancel
thf(fact_5799_ent__pure__pre__iff,axiom,
    ! [P2: assn,B: $o,Q2: assn] :
      ( ( entails @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ Q2 )
      = ( B
       => ( entails @ P2 @ Q2 ) ) ) ).

% ent_pure_pre_iff
thf(fact_5800_mod__pure__star__dist,axiom,
    ! [P2: assn,B: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ H2 )
      = ( ( rep_assn @ P2 @ H2 )
        & B ) ) ).

% mod_pure_star_dist
thf(fact_5801_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q2: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q2 )
      = ( B
       => ( entails @ one_one_assn @ Q2 ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_5802_ent__pure__post__iff,axiom,
    ! [P2: assn,Q2: assn,B: $o] :
      ( ( entails @ P2 @ ( times_times_assn @ Q2 @ ( pure_assn @ B ) ) )
      = ( ! [H4: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P2 @ H4 )
           => B )
        & ( entails @ P2 @ Q2 ) ) ) ).

% ent_pure_post_iff
thf(fact_5803_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_5804_ent__pure__post__iff__sng,axiom,
    ! [P2: assn,B: $o] :
      ( ( entails @ P2 @ ( pure_assn @ B ) )
      = ( ! [H4: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P2 @ H4 )
           => B )
        & ( entails @ P2 @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_5805_distinct__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( distinct_int @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( distinct_int @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_5806_distinct__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( distinct_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ ( nth_VEBT_VEBTi @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBTi @ Xs @ I ) ) )
          = ( distinct_VEBT_VEBTi @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_5807_distinct__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( distinct_VEBT_VEBT @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_5808_distinct__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
       => ( ( distinct_real @ ( list_update_real @ ( list_update_real @ Xs @ I @ ( nth_real @ Xs @ J ) ) @ J @ ( nth_real @ Xs @ I ) ) )
          = ( distinct_real @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_5809_distinct__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
       => ( ( distinct_o @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
          = ( distinct_o @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_5810_distinct__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_5811_VEBT__internal_OperInsTrans_Ointros_I2_J,axiom,
    ! [T: vEBT_VEBT,S: vEBT_VEBT,X: nat,U: vEBT_VEBT] :
      ( ( T
        = ( vEBT_vebt_insert @ S @ X ) )
     => ( ( vEBT_V6289311342943941716sTrans @ T @ U )
       => ( vEBT_V6289311342943941716sTrans @ S @ U ) ) ) ).

% VEBT_internal.perInsTrans.intros(2)
thf(fact_5812_VEBT__internal_OperInsTrans_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: vEBT_VEBT] :
      ( ( vEBT_V6289311342943941716sTrans @ A12 @ A23 )
     => ( ( A23 != A12 )
       => ~ ! [T6: vEBT_VEBT] :
              ( ? [X3: nat] :
                  ( T6
                  = ( vEBT_vebt_insert @ A12 @ X3 ) )
             => ~ ( vEBT_V6289311342943941716sTrans @ T6 @ A23 ) ) ) ) ).

% VEBT_internal.perInsTrans.cases
thf(fact_5813_VEBT__internal_OperInsTrans_Osimps,axiom,
    ( vEBT_V6289311342943941716sTrans
    = ( ^ [A13: vEBT_VEBT,A24: vEBT_VEBT] :
          ( ? [T2: vEBT_VEBT] :
              ( ( A13 = T2 )
              & ( A24 = T2 ) )
          | ? [T2: vEBT_VEBT,S8: vEBT_VEBT,X4: nat,U2: vEBT_VEBT] :
              ( ( A13 = S8 )
              & ( A24 = U2 )
              & ( T2
                = ( vEBT_vebt_insert @ S8 @ X4 ) )
              & ( vEBT_V6289311342943941716sTrans @ T2 @ U2 ) ) ) ) ) ).

% VEBT_internal.perInsTrans.simps
thf(fact_5814_set__take__subset,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ N @ Xs ) ) @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% set_take_subset
thf(fact_5815_set__take__subset,axiom,
    ! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% set_take_subset
thf(fact_5816_set__take__subset,axiom,
    ! [N: nat,Xs: list_real] : ( ord_less_eq_set_real @ ( set_real2 @ ( take_real @ N @ Xs ) ) @ ( set_real2 @ Xs ) ) ).

% set_take_subset
thf(fact_5817_set__take__subset,axiom,
    ! [N: nat,Xs: list_o] : ( ord_less_eq_set_o @ ( set_o2 @ ( take_o @ N @ Xs ) ) @ ( set_o2 @ Xs ) ) ).

% set_take_subset
thf(fact_5818_set__take__subset,axiom,
    ! [N: nat,Xs: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( take_int @ N @ Xs ) ) @ ( set_int2 @ Xs ) ) ).

% set_take_subset
thf(fact_5819_sorted__take,axiom,
    ! [Xs: list_rat,N: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ ( take_rat @ N @ Xs ) ) ) ).

% sorted_take
thf(fact_5820_sorted__take,axiom,
    ! [Xs: list_num,N: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
     => ( sorted_wrt_num @ ord_less_eq_num @ ( take_num @ N @ Xs ) ) ) ).

% sorted_take
thf(fact_5821_sorted__take,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs ) ) ) ).

% sorted_take
thf(fact_5822_sorted__take,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( take_int @ N @ Xs ) ) ) ).

% sorted_take
thf(fact_5823_finite__distinct__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( ( set_VEBT_VEBT2 @ Xs3 )
            = A4 )
          & ( distinct_VEBT_VEBT @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_5824_finite__distinct__list,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ? [Xs3: list_real] :
          ( ( ( set_real2 @ Xs3 )
            = A4 )
          & ( distinct_real @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_5825_finite__distinct__list,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ? [Xs3: list_o] :
          ( ( ( set_o2 @ Xs3 )
            = A4 )
          & ( distinct_o @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_5826_finite__distinct__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs3: list_nat] :
          ( ( ( set_nat2 @ Xs3 )
            = A4 )
          & ( distinct_nat @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_5827_finite__distinct__list,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ? [Xs3: list_int] :
          ( ( ( set_int2 @ Xs3 )
            = A4 )
          & ( distinct_int @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_5828_finite__distinct__list,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ? [Xs3: list_complex] :
          ( ( ( set_complex2 @ Xs3 )
            = A4 )
          & ( distinct_complex @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_5829_finite__distinct__list,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ? [Xs3: list_Code_integer] :
          ( ( ( set_Code_integer2 @ Xs3 )
            = A4 )
          & ( distin1543349897113766820nteger @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_5830_finite__set__image,axiom,
    ! [A4: set_list_VEBT_VEBT] :
      ( ( finite9070685783930458813T_VEBT @ ( image_6463372868993444447T_VEBT @ set_VEBT_VEBT2 @ A4 ) )
     => ( ! [Xs3: list_VEBT_VEBT] :
            ( ( member2936631157270082147T_VEBT @ Xs3 @ A4 )
           => ( distinct_VEBT_VEBT @ Xs3 ) )
       => ( finite3004134309566078307T_VEBT @ A4 ) ) ) ).

% finite_set_image
thf(fact_5831_finite__set__image,axiom,
    ! [A4: set_list_nat] :
      ( ( finite1152437895449049373et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ A4 ) )
     => ( ! [Xs3: list_nat] :
            ( ( member_list_nat @ Xs3 @ A4 )
           => ( distinct_nat @ Xs3 ) )
       => ( finite8100373058378681591st_nat @ A4 ) ) ) ).

% finite_set_image
thf(fact_5832_finite__set__image,axiom,
    ! [A4: set_list_real] :
      ( ( finite9007344921179782393t_real @ ( image_6239767680843085477t_real @ set_real2 @ A4 ) )
     => ( ! [Xs3: list_real] :
            ( ( member_list_real @ Xs3 @ A4 )
           => ( distinct_real @ Xs3 ) )
       => ( finite306553202115118035t_real @ A4 ) ) ) ).

% finite_set_image
thf(fact_5833_finite__set__image,axiom,
    ! [A4: set_list_o] :
      ( ( finite_finite_set_o @ ( image_list_o_set_o @ set_o2 @ A4 ) )
     => ( ! [Xs3: list_o] :
            ( ( member_list_o @ Xs3 @ A4 )
           => ( distinct_o @ Xs3 ) )
       => ( finite_finite_list_o @ A4 ) ) ) ).

% finite_set_image
thf(fact_5834_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ M @ Xs ) ) @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_5835_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_5836_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_real @ ( set_real2 @ ( take_real @ M @ Xs ) ) @ ( set_real2 @ ( take_real @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_5837_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_o @ ( set_o2 @ ( take_o @ M @ Xs ) ) @ ( set_o2 @ ( take_o @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_5838_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_int @ ( set_int2 @ ( take_int @ M @ Xs ) ) @ ( set_int2 @ ( take_int @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_5839_strict__sorted__iff,axiom,
    ! [L: list_real] :
      ( ( sorted_wrt_real @ ord_less_real @ L )
      = ( ( sorted_wrt_real @ ord_less_eq_real @ L )
        & ( distinct_real @ L ) ) ) ).

% strict_sorted_iff
thf(fact_5840_strict__sorted__iff,axiom,
    ! [L: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ L )
      = ( ( sorted_wrt_rat @ ord_less_eq_rat @ L )
        & ( distinct_rat @ L ) ) ) ).

% strict_sorted_iff
thf(fact_5841_strict__sorted__iff,axiom,
    ! [L: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ L )
      = ( ( sorted_wrt_num @ ord_less_eq_num @ L )
        & ( distinct_num @ L ) ) ) ).

% strict_sorted_iff
thf(fact_5842_strict__sorted__iff,axiom,
    ! [L: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ L )
      = ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
        & ( distinct_nat @ L ) ) ) ).

% strict_sorted_iff
thf(fact_5843_strict__sorted__iff,axiom,
    ! [L: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ L )
      = ( ( sorted_wrt_int @ ord_less_eq_int @ L )
        & ( distinct_int @ L ) ) ) ).

% strict_sorted_iff
thf(fact_5844_distinct__conv__nth,axiom,
    ( distinct_VEBT_VEBTi
    = ( ^ [Xs2: list_VEBT_VEBTi] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
             => ( ( I3 != J3 )
               => ( ( nth_VEBT_VEBTi @ Xs2 @ I3 )
                 != ( nth_VEBT_VEBTi @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_5845_distinct__conv__nth,axiom,
    ( distinct_int
    = ( ^ [Xs2: list_int] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
             => ( ( I3 != J3 )
               => ( ( nth_int @ Xs2 @ I3 )
                 != ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_5846_distinct__conv__nth,axiom,
    ( distinct_VEBT_VEBT
    = ( ^ [Xs2: list_VEBT_VEBT] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( I3 != J3 )
               => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
                 != ( nth_VEBT_VEBT @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_5847_distinct__conv__nth,axiom,
    ( distinct_real
    = ( ^ [Xs2: list_real] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs2 ) )
             => ( ( I3 != J3 )
               => ( ( nth_real @ Xs2 @ I3 )
                 != ( nth_real @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_5848_distinct__conv__nth,axiom,
    ( distinct_o
    = ( ^ [Xs2: list_o] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs2 ) )
             => ( ( I3 != J3 )
               => ( ( nth_o @ Xs2 @ I3 )
                 != ( nth_o @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_5849_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs2: list_nat] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( I3 != J3 )
               => ( ( nth_nat @ Xs2 @ I3 )
                 != ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_5850_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,J: nat] :
      ( ( distinct_VEBT_VEBTi @ Xs )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs ) )
         => ( ( ( nth_VEBT_VEBTi @ Xs @ I )
              = ( nth_VEBT_VEBTi @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_5851_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( distinct_int @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ( ( nth_int @ Xs @ I )
              = ( nth_int @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_5852_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( ( ( nth_VEBT_VEBT @ Xs @ I )
              = ( nth_VEBT_VEBT @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_5853_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_real,I: nat,J: nat] :
      ( ( distinct_real @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
         => ( ( ( nth_real @ Xs @ I )
              = ( nth_real @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_5854_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_o,I: nat,J: nat] :
      ( ( distinct_o @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
         => ( ( ( nth_o @ Xs @ I )
              = ( nth_o @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_5855_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_5856_distinct__length__le,axiom,
    ! [Ys: list_VEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Ys )
     => ( ( ( set_VEBT_VEBT2 @ Ys )
          = ( set_VEBT_VEBT2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_5857_distinct__length__le,axiom,
    ! [Ys: list_real,Xs: list_real] :
      ( ( distinct_real @ Ys )
     => ( ( ( set_real2 @ Ys )
          = ( set_real2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_real @ Ys ) @ ( size_size_list_real @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_5858_distinct__length__le,axiom,
    ! [Ys: list_o,Xs: list_o] :
      ( ( distinct_o @ Ys )
     => ( ( ( set_o2 @ Ys )
          = ( set_o2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_5859_distinct__length__le,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( distinct_nat @ Ys )
     => ( ( ( set_nat2 @ Ys )
          = ( set_nat2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_5860_sorted__distinct__set__unique,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
     => ( ( distinct_real @ Xs )
       => ( ( sorted_wrt_real @ ord_less_eq_real @ Ys )
         => ( ( distinct_real @ Ys )
           => ( ( ( set_real2 @ Xs )
                = ( set_real2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_5861_sorted__distinct__set__unique,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
     => ( ( distinct_o @ Xs )
       => ( ( sorted_wrt_o @ ord_less_eq_o @ Ys )
         => ( ( distinct_o @ Ys )
           => ( ( ( set_o2 @ Xs )
                = ( set_o2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_5862_sorted__distinct__set__unique,axiom,
    ! [Xs: list_rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
     => ( ( distinct_rat @ Xs )
       => ( ( sorted_wrt_rat @ ord_less_eq_rat @ Ys )
         => ( ( distinct_rat @ Ys )
           => ( ( ( set_rat2 @ Xs )
                = ( set_rat2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_5863_sorted__distinct__set__unique,axiom,
    ! [Xs: list_num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
     => ( ( distinct_num @ Xs )
       => ( ( sorted_wrt_num @ ord_less_eq_num @ Ys )
         => ( ( distinct_num @ Ys )
           => ( ( ( set_num2 @ Xs )
                = ( set_num2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_5864_sorted__distinct__set__unique,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( distinct_nat @ Xs )
       => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
         => ( ( distinct_nat @ Ys )
           => ( ( ( set_nat2 @ Xs )
                = ( set_nat2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_5865_sorted__distinct__set__unique,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( distinct_int @ Xs )
       => ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
         => ( ( distinct_int @ Ys )
           => ( ( ( set_int2 @ Xs )
                = ( set_int2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_5866_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X ) ) ).

% VEBT_internal.insert'.simps(1)
thf(fact_5867_vebt__insert__code_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert_code(1)
thf(fact_5868_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_eq_nat @ K @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s7982070591426661849_VEBTi @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                = ( nth_VEBT_VEBTi @ Ys @ I2 ) ) )
         => ( ( take_VEBT_VEBTi @ K @ Xs )
            = ( take_VEBT_VEBTi @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_5869_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_int @ Xs @ I2 )
                = ( nth_int @ Ys @ I2 ) ) )
         => ( ( take_int @ K @ Xs )
            = ( take_int @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_5870_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ K @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
                = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
         => ( ( take_VEBT_VEBT @ K @ Xs )
            = ( take_VEBT_VEBT @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_5871_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_real,Ys: list_real] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_real @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_real @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_real @ Xs @ I2 )
                = ( nth_real @ Ys @ I2 ) ) )
         => ( ( take_real @ K @ Xs )
            = ( take_real @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_5872_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_o,Ys: list_o] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_o @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_o @ Xs @ I2 )
                = ( nth_o @ Ys @ I2 ) ) )
         => ( ( take_o @ K @ Xs )
            = ( take_o @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_5873_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_nat @ Xs @ I2 )
                = ( nth_nat @ Ys @ I2 ) ) )
         => ( ( take_nat @ K @ Xs )
            = ( take_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_5874_finite__sorted__distinct__unique,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ? [X3: list_real] :
          ( ( ( set_real2 @ X3 )
            = A4 )
          & ( sorted_wrt_real @ ord_less_eq_real @ X3 )
          & ( distinct_real @ X3 )
          & ! [Y5: list_real] :
              ( ( ( ( set_real2 @ Y5 )
                  = A4 )
                & ( sorted_wrt_real @ ord_less_eq_real @ Y5 )
                & ( distinct_real @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_5875_finite__sorted__distinct__unique,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ? [X3: list_o] :
          ( ( ( set_o2 @ X3 )
            = A4 )
          & ( sorted_wrt_o @ ord_less_eq_o @ X3 )
          & ( distinct_o @ X3 )
          & ! [Y5: list_o] :
              ( ( ( ( set_o2 @ Y5 )
                  = A4 )
                & ( sorted_wrt_o @ ord_less_eq_o @ Y5 )
                & ( distinct_o @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_5876_finite__sorted__distinct__unique,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ? [X3: list_Code_integer] :
          ( ( ( set_Code_integer2 @ X3 )
            = A4 )
          & ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ X3 )
          & ( distin1543349897113766820nteger @ X3 )
          & ! [Y5: list_Code_integer] :
              ( ( ( ( set_Code_integer2 @ Y5 )
                  = A4 )
                & ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ Y5 )
                & ( distin1543349897113766820nteger @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_5877_finite__sorted__distinct__unique,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ? [X3: list_rat] :
          ( ( ( set_rat2 @ X3 )
            = A4 )
          & ( sorted_wrt_rat @ ord_less_eq_rat @ X3 )
          & ( distinct_rat @ X3 )
          & ! [Y5: list_rat] :
              ( ( ( ( set_rat2 @ Y5 )
                  = A4 )
                & ( sorted_wrt_rat @ ord_less_eq_rat @ Y5 )
                & ( distinct_rat @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_5878_finite__sorted__distinct__unique,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ? [X3: list_num] :
          ( ( ( set_num2 @ X3 )
            = A4 )
          & ( sorted_wrt_num @ ord_less_eq_num @ X3 )
          & ( distinct_num @ X3 )
          & ! [Y5: list_num] :
              ( ( ( ( set_num2 @ Y5 )
                  = A4 )
                & ( sorted_wrt_num @ ord_less_eq_num @ Y5 )
                & ( distinct_num @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_5879_finite__sorted__distinct__unique,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [X3: list_nat] :
          ( ( ( set_nat2 @ X3 )
            = A4 )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ X3 )
          & ( distinct_nat @ X3 )
          & ! [Y5: list_nat] :
              ( ( ( ( set_nat2 @ Y5 )
                  = A4 )
                & ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
                & ( distinct_nat @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_5880_finite__sorted__distinct__unique,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ? [X3: list_int] :
          ( ( ( set_int2 @ X3 )
            = A4 )
          & ( sorted_wrt_int @ ord_less_eq_int @ X3 )
          & ( distinct_int @ X3 )
          & ! [Y5: list_int] :
              ( ( ( ( set_int2 @ Y5 )
                  = A4 )
                & ( sorted_wrt_int @ ord_less_eq_int @ Y5 )
                & ( distinct_int @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_5881_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( distinct_VEBT_VEBTi @ Xs )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
            & ( ( nth_VEBT_VEBTi @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                  & ( ( nth_VEBT_VEBTi @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_5882_distinct__Ex1,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_int @ Xs ) )
                  & ( ( nth_int @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_5883_distinct__Ex1,axiom,
    ! [Xs: list_set_nat,X: set_nat] :
      ( ( distinct_set_nat @ Xs )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s3254054031482475050et_nat @ Xs ) )
                  & ( ( nth_set_nat @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_5884_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                  & ( ( nth_VEBT_VEBT @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_5885_distinct__Ex1,axiom,
    ! [Xs: list_real,X: real] :
      ( ( distinct_real @ Xs )
     => ( ( member_real @ X @ ( set_real2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_real @ Xs ) )
                  & ( ( nth_real @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_5886_distinct__Ex1,axiom,
    ! [Xs: list_o,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( member_o @ X @ ( set_o2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_o @ Xs ) )
                  & ( ( nth_o @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_5887_distinct__Ex1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_5888_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
     => ( ( distinct_real @ Xs )
       => ( ( linord4252657396651189596t_real @ ( set_real2 @ Xs ) )
          = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5889_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
     => ( ( distinct_o @ Xs )
       => ( ( linord3142498349692569832_set_o @ ( set_o2 @ Xs ) )
          = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5890_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
     => ( ( distinct_rat @ Xs )
       => ( ( linord1979837681955606664et_rat @ ( set_rat2 @ Xs ) )
          = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5891_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
     => ( ( distinct_num @ Xs )
       => ( ( linord8395671565052656842et_num @ ( set_num2 @ Xs ) )
          = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5892_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( distinct_int @ Xs )
       => ( ( linord2612477271533052124et_int @ ( set_int2 @ Xs ) )
          = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5893_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( distinct_nat @ Xs )
       => ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
          = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5894_distinct__sorted__mono,axiom,
    ! [L: list_o,I: nat,J: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ L )
     => ( ( distinct_o @ L )
       => ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_o @ L ) )
           => ( ord_less_o @ ( nth_o @ L @ I ) @ ( nth_o @ L @ J ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_5895_distinct__sorted__mono,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ L )
     => ( ( distinct_real @ L )
       => ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_real @ L ) )
           => ( ord_less_real @ ( nth_real @ L @ I ) @ ( nth_real @ L @ J ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_5896_distinct__sorted__mono,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ L )
     => ( ( distinct_rat @ L )
       => ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_rat @ L ) )
           => ( ord_less_rat @ ( nth_rat @ L @ I ) @ ( nth_rat @ L @ J ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_5897_distinct__sorted__mono,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ L )
     => ( ( distinct_num @ L )
       => ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_num @ L ) )
           => ( ord_less_num @ ( nth_num @ L @ I ) @ ( nth_num @ L @ J ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_5898_distinct__sorted__mono,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
     => ( ( distinct_nat @ L )
       => ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
           => ( ord_less_nat @ ( nth_nat @ L @ I ) @ ( nth_nat @ L @ J ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_5899_distinct__sorted__mono,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ L )
     => ( ( distinct_int @ L )
       => ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_int @ L ) )
           => ( ord_less_int @ ( nth_int @ L @ I ) @ ( nth_int @ L @ J ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_5900_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_o,I: nat,J: nat] :
      ( ( distinct_o @ L )
     => ( ( sorted_wrt_o @ ord_less_eq_o @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_o @ L ) )
           => ( ( ord_less_o @ ( nth_o @ L @ I ) @ ( nth_o @ L @ J ) )
              = ( ord_less_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_5901_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( ( distinct_real @ L )
     => ( ( sorted_wrt_real @ ord_less_eq_real @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_real @ L ) )
           => ( ( ord_less_real @ ( nth_real @ L @ I ) @ ( nth_real @ L @ J ) )
              = ( ord_less_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_5902_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( ( distinct_rat @ L )
     => ( ( sorted_wrt_rat @ ord_less_eq_rat @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_rat @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_rat @ L ) )
           => ( ( ord_less_rat @ ( nth_rat @ L @ I ) @ ( nth_rat @ L @ J ) )
              = ( ord_less_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_5903_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( ( distinct_num @ L )
     => ( ( sorted_wrt_num @ ord_less_eq_num @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_num @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_num @ L ) )
           => ( ( ord_less_num @ ( nth_num @ L @ I ) @ ( nth_num @ L @ J ) )
              = ( ord_less_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_5904_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ L )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
           => ( ( ord_less_nat @ ( nth_nat @ L @ I ) @ ( nth_nat @ L @ J ) )
              = ( ord_less_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_5905_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( ( distinct_int @ L )
     => ( ( sorted_wrt_int @ ord_less_eq_int @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_int @ L ) )
           => ( ( ord_less_int @ ( nth_int @ L @ I ) @ ( nth_int @ L @ J ) )
              = ( ord_less_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_5906_distinct__sorted__mono__iff,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( ( distinct_real @ L )
     => ( ( sorted_wrt_real @ ord_less_eq_real @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_real @ L ) )
           => ( ( ord_less_eq_real @ ( nth_real @ L @ I ) @ ( nth_real @ L @ J ) )
              = ( ord_less_eq_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
thf(fact_5907_distinct__sorted__mono__iff,axiom,
    ! [L: list_o,I: nat,J: nat] :
      ( ( distinct_o @ L )
     => ( ( sorted_wrt_o @ ord_less_eq_o @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_o @ L ) )
           => ( ( ord_less_eq_o @ ( nth_o @ L @ I ) @ ( nth_o @ L @ J ) )
              = ( ord_less_eq_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
thf(fact_5908_distinct__sorted__mono__iff,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( ( distinct_rat @ L )
     => ( ( sorted_wrt_rat @ ord_less_eq_rat @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_rat @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_rat @ L ) )
           => ( ( ord_less_eq_rat @ ( nth_rat @ L @ I ) @ ( nth_rat @ L @ J ) )
              = ( ord_less_eq_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
thf(fact_5909_distinct__sorted__mono__iff,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( ( distinct_num @ L )
     => ( ( sorted_wrt_num @ ord_less_eq_num @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_num @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_num @ L ) )
           => ( ( ord_less_eq_num @ ( nth_num @ L @ I ) @ ( nth_num @ L @ J ) )
              = ( ord_less_eq_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
thf(fact_5910_distinct__sorted__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ L )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
           => ( ( ord_less_eq_nat @ ( nth_nat @ L @ I ) @ ( nth_nat @ L @ J ) )
              = ( ord_less_eq_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
thf(fact_5911_distinct__sorted__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( ( distinct_int @ L )
     => ( ( sorted_wrt_int @ ord_less_eq_int @ L )
       => ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_int @ L ) )
           => ( ( ord_less_eq_int @ ( nth_int @ L @ I ) @ ( nth_int @ L @ J ) )
              = ( ord_less_eq_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
thf(fact_5912_distinct__list__update,axiom,
    ! [Xs: list_set_nat,A: set_nat,I: nat] :
      ( ( distinct_set_nat @ Xs )
     => ( ~ ( member_set_nat @ A @ ( minus_2163939370556025621et_nat @ ( set_set_nat2 @ Xs ) @ ( insert_set_nat @ ( nth_set_nat @ Xs @ I ) @ bot_bot_set_set_nat ) ) )
       => ( distinct_set_nat @ ( list_update_set_nat @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_5913_distinct__list__update,axiom,
    ! [Xs: list_real,A: real,I: nat] :
      ( ( distinct_real @ Xs )
     => ( ~ ( member_real @ A @ ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ ( nth_real @ Xs @ I ) @ bot_bot_set_real ) ) )
       => ( distinct_real @ ( list_update_real @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_5914_distinct__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,A: vEBT_VEBTi,I: nat] :
      ( ( distinct_VEBT_VEBTi @ Xs )
     => ( ~ ( member_VEBT_VEBTi @ A @ ( minus_3697805406911847364_VEBTi @ ( set_VEBT_VEBTi2 @ Xs ) @ ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ I ) @ bot_bo8982466882572371071_VEBTi ) ) )
       => ( distinct_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_5915_distinct__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,A: vEBT_VEBT,I: nat] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ~ ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ bot_bo8194388402131092736T_VEBT ) ) )
       => ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_5916_distinct__list__update,axiom,
    ! [Xs: list_int,A: int,I: nat] :
      ( ( distinct_int @ Xs )
     => ( ~ ( member_int @ A @ ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ ( nth_int @ Xs @ I ) @ bot_bot_set_int ) ) )
       => ( distinct_int @ ( list_update_int @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_5917_distinct__list__update,axiom,
    ! [Xs: list_o,A: $o,I: nat] :
      ( ( distinct_o @ Xs )
     => ( ~ ( member_o @ A @ ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ ( nth_o @ Xs @ I ) @ bot_bot_set_o ) ) )
       => ( distinct_o @ ( list_update_o @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_5918_distinct__list__update,axiom,
    ! [Xs: list_nat,A: nat,I: nat] :
      ( ( distinct_nat @ Xs )
     => ( ~ ( member_nat @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ I ) @ bot_bot_set_nat ) ) )
       => ( distinct_nat @ ( list_update_nat @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_5919_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_5920_heaphelp,axiom,
    ! [Xa2: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N: 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 @ Xa2 @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                & ( N = N ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N @ Xa2 @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_5921_heaphelp,axiom,
    ! [Xa2: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N: 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 @ Xa2 @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_nat = none_nat )
                & ( N = N ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N @ Xa2 @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_5922_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_5923_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_5924_mergesort__remdups__correct,axiom,
    ! [L: list_real] :
      ( ( distinct_real @ ( merges7559785487730971421s_real @ L ) )
      & ( sorted_wrt_real @ ord_less_eq_real @ ( merges7559785487730971421s_real @ L ) )
      & ( ( set_real2 @ ( merges7559785487730971421s_real @ L ) )
        = ( set_real2 @ L ) ) ) ).

% mergesort_remdups_correct
thf(fact_5925_mergesort__remdups__correct,axiom,
    ! [L: list_o] :
      ( ( distinct_o @ ( mergesort_remdups_o @ L ) )
      & ( sorted_wrt_o @ ord_less_eq_o @ ( mergesort_remdups_o @ L ) )
      & ( ( set_o2 @ ( mergesort_remdups_o @ L ) )
        = ( set_o2 @ L ) ) ) ).

% mergesort_remdups_correct
thf(fact_5926_mergesort__remdups__correct,axiom,
    ! [L: list_rat] :
      ( ( distinct_rat @ ( merges1021483306759835337ps_rat @ L ) )
      & ( sorted_wrt_rat @ ord_less_eq_rat @ ( merges1021483306759835337ps_rat @ L ) )
      & ( ( set_rat2 @ ( merges1021483306759835337ps_rat @ L ) )
        = ( set_rat2 @ L ) ) ) ).

% mergesort_remdups_correct
thf(fact_5927_mergesort__remdups__correct,axiom,
    ! [L: list_num] :
      ( ( distinct_num @ ( merges7437317189856885515ps_num @ L ) )
      & ( sorted_wrt_num @ ord_less_eq_num @ ( merges7437317189856885515ps_num @ L ) )
      & ( ( set_num2 @ ( merges7437317189856885515ps_num @ L ) )
        = ( set_num2 @ L ) ) ) ).

% mergesort_remdups_correct
thf(fact_5928_mergesort__remdups__correct,axiom,
    ! [L: list_nat] :
      ( ( distinct_nat @ ( merges1656613366846331073ps_nat @ L ) )
      & ( sorted_wrt_nat @ ord_less_eq_nat @ ( merges1656613366846331073ps_nat @ L ) )
      & ( ( set_nat2 @ ( merges1656613366846331073ps_nat @ L ) )
        = ( set_nat2 @ L ) ) ) ).

% mergesort_remdups_correct
thf(fact_5929_mergesort__remdups__correct,axiom,
    ! [L: list_int] :
      ( ( distinct_int @ ( merges1654122896337280797ps_int @ L ) )
      & ( sorted_wrt_int @ ord_less_eq_int @ ( merges1654122896337280797ps_int @ L ) )
      & ( ( set_int2 @ ( merges1654122896337280797ps_int @ L ) )
        = ( set_int2 @ L ) ) ) ).

% mergesort_remdups_correct
thf(fact_5930_last__take__nth__conv,axiom,
    ! [N: nat,L: list_VEBT_VEBTi] :
      ( ( ord_less_eq_nat @ N @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( N != zero_zero_nat )
       => ( ( last_VEBT_VEBTi @ ( take_VEBT_VEBTi @ N @ L ) )
          = ( nth_VEBT_VEBTi @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% last_take_nth_conv
thf(fact_5931_last__take__nth__conv,axiom,
    ! [N: nat,L: list_int] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ L ) )
     => ( ( N != zero_zero_nat )
       => ( ( last_int @ ( take_int @ N @ L ) )
          = ( nth_int @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% last_take_nth_conv
thf(fact_5932_last__take__nth__conv,axiom,
    ! [N: nat,L: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( N != zero_zero_nat )
       => ( ( last_VEBT_VEBT @ ( take_VEBT_VEBT @ N @ L ) )
          = ( nth_VEBT_VEBT @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% last_take_nth_conv
thf(fact_5933_last__take__nth__conv,axiom,
    ! [N: nat,L: list_real] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ L ) )
     => ( ( N != zero_zero_nat )
       => ( ( last_real @ ( take_real @ N @ L ) )
          = ( nth_real @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% last_take_nth_conv
thf(fact_5934_last__take__nth__conv,axiom,
    ! [N: nat,L: list_o] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ L ) )
     => ( ( N != zero_zero_nat )
       => ( ( last_o @ ( take_o @ N @ L ) )
          = ( nth_o @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% last_take_nth_conv
thf(fact_5935_last__take__nth__conv,axiom,
    ! [N: nat,L: list_nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ L ) )
     => ( ( N != zero_zero_nat )
       => ( ( last_nat @ ( take_nat @ N @ L ) )
          = ( nth_nat @ L @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% last_take_nth_conv
thf(fact_5936_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_5937_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd2: $o,Ve2: $o,V2: option4927543243414619207at_nat,Va2: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd2 @ Ve2 ) @ ( vEBT_Nodei @ V2 @ Va2 @ Vb @ Vc ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(4)
thf(fact_5938_subset__subseqs,axiom,
    ! [X6: set_VEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ X6 @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( member_set_VEBT_VEBT @ X6 @ ( image_6463372868993444447T_VEBT @ set_VEBT_VEBT2 @ ( set_list_VEBT_VEBT2 @ ( subseqs_VEBT_VEBT @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_5939_subset__subseqs,axiom,
    ! [X6: set_nat,Xs: list_nat] :
      ( ( ord_less_eq_set_nat @ X6 @ ( set_nat2 @ Xs ) )
     => ( member_set_nat @ X6 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_5940_subset__subseqs,axiom,
    ! [X6: set_real,Xs: list_real] :
      ( ( ord_less_eq_set_real @ X6 @ ( set_real2 @ Xs ) )
     => ( member_set_real @ X6 @ ( image_6239767680843085477t_real @ set_real2 @ ( set_list_real2 @ ( subseqs_real @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_5941_subset__subseqs,axiom,
    ! [X6: set_o,Xs: list_o] :
      ( ( ord_less_eq_set_o @ X6 @ ( set_o2 @ Xs ) )
     => ( member_set_o @ X6 @ ( image_list_o_set_o @ set_o2 @ ( set_list_o2 @ ( subseqs_o @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_5942_subset__subseqs,axiom,
    ! [X6: set_int,Xs: list_int] :
      ( ( ord_less_eq_set_int @ X6 @ ( set_int2 @ Xs ) )
     => ( member_set_int @ X6 @ ( image_3606813740839090725et_int @ set_int2 @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_5943_set__remove1__eq,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ X @ Xs ) )
        = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_remove1_eq
thf(fact_5944_set__remove1__eq,axiom,
    ! [Xs: list_real,X: real] :
      ( ( distinct_real @ Xs )
     => ( ( set_real2 @ ( remove1_real @ X @ Xs ) )
        = ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_remove1_eq
thf(fact_5945_set__remove1__eq,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( set_int2 @ ( remove1_int @ X @ Xs ) )
        = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_remove1_eq
thf(fact_5946_set__remove1__eq,axiom,
    ! [Xs: list_o,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( set_o2 @ ( remove1_o @ X @ Xs ) )
        = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% set_remove1_eq
thf(fact_5947_set__remove1__eq,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( set_nat2 @ ( remove1_nat @ X @ Xs ) )
        = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_5948_lex__take__index,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi,R: set_Pr2227491710730465451_VEBTi] :
      ( ( member4173000155140927252_VEBTi @ ( produc4384243565435462691_VEBTi @ Xs @ Ys ) @ ( lex_VEBT_VEBTi @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Ys ) )
             => ( ( ( take_VEBT_VEBTi @ I2 @ Xs )
                  = ( take_VEBT_VEBTi @ I2 @ Ys ) )
               => ~ ( member660371905731732212_VEBTi @ ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ I2 ) @ ( nth_VEBT_VEBTi @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_5949_lex__take__index,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Xs @ Ys ) @ ( lex_uint32 @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s4844771616002835472uint32 @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s4844771616002835472uint32 @ Ys ) )
             => ( ( ( take_uint32 @ I2 @ Xs )
                  = ( take_uint32 @ I2 @ Ys ) )
               => ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ ( nth_uint32 @ Xs @ I2 ) @ ( nth_uint32 @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_5950_lex__take__index,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys ) )
             => ( ( ( take_int @ I2 @ Xs )
                  = ( take_int @ I2 @ Ys ) )
               => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_5951_lex__take__index,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( lex_VEBT_VEBT @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Ys ) )
             => ( ( ( take_VEBT_VEBT @ I2 @ Xs )
                  = ( take_VEBT_VEBT @ I2 @ Ys ) )
               => ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I2 ) @ ( nth_VEBT_VEBT @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_5952_lex__take__index,axiom,
    ! [Xs: list_real,Ys: list_real,R: set_Pr6218003697084177305l_real] :
      ( ( member6584958104391596930t_real @ ( produc1408950526243324945t_real @ Xs @ Ys ) @ ( lex_real @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Ys ) )
             => ( ( ( take_real @ I2 @ Xs )
                  = ( take_real @ I2 @ Ys ) )
               => ~ ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ ( nth_real @ Xs @ I2 ) @ ( nth_real @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_5953_lex__take__index,axiom,
    ! [Xs: list_o,Ys: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( lex_o @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys ) )
             => ( ( ( take_o @ I2 @ Xs )
                  = ( take_o @ I2 @ Ys ) )
               => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ I2 ) @ ( nth_o @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_5954_lex__take__index,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
             => ( ( ( take_nat @ I2 @ Xs )
                  = ( take_nat @ I2 @ Ys ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_5955_take__butlast__conv,axiom,
    ! [L: list_VEBT_VEBT] :
      ( ( take_VEBT_VEBT @ ( minus_minus_nat @ ( size_s6755466524823107622T_VEBT @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
      = ( butlast_VEBT_VEBT @ L ) ) ).

% take_butlast_conv
thf(fact_5956_take__butlast__conv,axiom,
    ! [L: list_real] :
      ( ( take_real @ ( minus_minus_nat @ ( size_size_list_real @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
      = ( butlast_real @ L ) ) ).

% take_butlast_conv
thf(fact_5957_take__butlast__conv,axiom,
    ! [L: list_o] :
      ( ( take_o @ ( minus_minus_nat @ ( size_size_list_o @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
      = ( butlast_o @ L ) ) ).

% take_butlast_conv
thf(fact_5958_take__butlast__conv,axiom,
    ! [L: list_nat] :
      ( ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
      = ( butlast_nat @ L ) ) ).

% take_butlast_conv
thf(fact_5959_take__minus__one__conv__butlast,axiom,
    ! [N: nat,L: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( take_VEBT_VEBT @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ L )
        = ( butlast_VEBT_VEBT @ ( take_VEBT_VEBT @ N @ L ) ) ) ) ).

% take_minus_one_conv_butlast
thf(fact_5960_take__minus__one__conv__butlast,axiom,
    ! [N: nat,L: list_real] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ L ) )
     => ( ( take_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ L )
        = ( butlast_real @ ( take_real @ N @ L ) ) ) ) ).

% take_minus_one_conv_butlast
thf(fact_5961_take__minus__one__conv__butlast,axiom,
    ! [N: nat,L: list_o] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ L ) )
     => ( ( take_o @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ L )
        = ( butlast_o @ ( take_o @ N @ L ) ) ) ) ).

% take_minus_one_conv_butlast
thf(fact_5962_take__minus__one__conv__butlast,axiom,
    ! [N: nat,L: list_nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ L ) )
     => ( ( take_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ L )
        = ( butlast_nat @ ( take_nat @ N @ L ) ) ) ) ).

% take_minus_one_conv_butlast
thf(fact_5963_Cons__in__lex,axiom,
    ! [X: uint32,Xs: list_uint32,Y: uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ ( cons_uint32 @ X @ Xs ) @ ( cons_uint32 @ Y @ Ys ) ) @ ( lex_uint32 @ R ) )
      = ( ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ R )
          & ( ( size_s4844771616002835472uint32 @ Xs )
            = ( size_s4844771616002835472uint32 @ Ys ) ) )
        | ( ( X = Y )
          & ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Xs @ Ys ) @ ( lex_uint32 @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_5964_Cons__in__lex,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( lex_int @ R ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
          & ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys ) ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_5965_Cons__in__lex,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( cons_VEBT_VEBT @ Y @ Ys ) ) @ ( lex_VEBT_VEBT @ R ) )
      = ( ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y ) @ R )
          & ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys ) ) )
        | ( ( X = Y )
          & ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( lex_VEBT_VEBT @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_5966_Cons__in__lex,axiom,
    ! [X: real,Xs: list_real,Y: real,Ys: list_real,R: set_Pr6218003697084177305l_real] :
      ( ( member6584958104391596930t_real @ ( produc1408950526243324945t_real @ ( cons_real @ X @ Xs ) @ ( cons_real @ Y @ Ys ) ) @ ( lex_real @ R ) )
      = ( ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ R )
          & ( ( size_size_list_real @ Xs )
            = ( size_size_list_real @ Ys ) ) )
        | ( ( X = Y )
          & ( member6584958104391596930t_real @ ( produc1408950526243324945t_real @ Xs @ Ys ) @ ( lex_real @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_5967_Cons__in__lex,axiom,
    ! [X: $o,Xs: list_o,Y: $o,Ys: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs ) @ ( cons_o @ Y @ Ys ) ) @ ( lex_o @ R ) )
      = ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
          & ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys ) ) )
        | ( ( X = Y )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( lex_o @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_5968_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_5969_butlast__update_H,axiom,
    ! [L: list_nat,I: nat,X: nat] :
      ( ( list_update_nat @ ( butlast_nat @ L ) @ I @ X )
      = ( butlast_nat @ ( list_update_nat @ L @ I @ X ) ) ) ).

% butlast_update'
thf(fact_5970_butlast__update_H,axiom,
    ! [L: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( butlast_VEBT_VEBTi @ L ) @ I @ X )
      = ( butlast_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X ) ) ) ).

% butlast_update'
thf(fact_5971_butlast__update_H,axiom,
    ! [L: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( butlast_VEBT_VEBT @ L ) @ I @ X )
      = ( butlast_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) ) ) ).

% butlast_update'
thf(fact_5972_foldl__absorb1,axiom,
    ! [X: rat,Zs: list_rat] :
      ( ( times_times_rat @ X @ ( foldl_rat_rat @ times_times_rat @ one_one_rat @ Zs ) )
      = ( foldl_rat_rat @ times_times_rat @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_5973_foldl__absorb1,axiom,
    ! [X: assn,Zs: list_assn] :
      ( ( times_times_assn @ X @ ( foldl_assn_assn @ times_times_assn @ one_one_assn @ Zs ) )
      = ( foldl_assn_assn @ times_times_assn @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_5974_foldl__absorb1,axiom,
    ! [X: uint32,Zs: list_uint32] :
      ( ( times_times_uint32 @ X @ ( foldl_uint32_uint32 @ times_times_uint32 @ one_one_uint32 @ Zs ) )
      = ( foldl_uint32_uint32 @ times_times_uint32 @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_5975_foldl__absorb1,axiom,
    ! [X: real,Zs: list_real] :
      ( ( times_times_real @ X @ ( foldl_real_real @ times_times_real @ one_one_real @ Zs ) )
      = ( foldl_real_real @ times_times_real @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_5976_foldl__absorb1,axiom,
    ! [X: nat,Zs: list_nat] :
      ( ( times_times_nat @ X @ ( foldl_nat_nat @ times_times_nat @ one_one_nat @ Zs ) )
      = ( foldl_nat_nat @ times_times_nat @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_5977_foldl__absorb1,axiom,
    ! [X: int,Zs: list_int] :
      ( ( times_times_int @ X @ ( foldl_int_int @ times_times_int @ one_one_int @ Zs ) )
      = ( foldl_int_int @ times_times_int @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_5978_set__remove1__subset,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ X @ Xs ) ) @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_5979_set__remove1__subset,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_5980_set__remove1__subset,axiom,
    ! [X: real,Xs: list_real] : ( ord_less_eq_set_real @ ( set_real2 @ ( remove1_real @ X @ Xs ) ) @ ( set_real2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_5981_set__remove1__subset,axiom,
    ! [X: $o,Xs: list_o] : ( ord_less_eq_set_o @ ( set_o2 @ ( remove1_o @ X @ Xs ) ) @ ( set_o2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_5982_set__remove1__subset,axiom,
    ! [X: int,Xs: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( remove1_int @ X @ Xs ) ) @ ( set_int2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_5983_sorted__remove1,axiom,
    ! [Xs: list_rat,A: rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ ( remove1_rat @ A @ Xs ) ) ) ).

% sorted_remove1
thf(fact_5984_sorted__remove1,axiom,
    ! [Xs: list_num,A: num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
     => ( sorted_wrt_num @ ord_less_eq_num @ ( remove1_num @ A @ Xs ) ) ) ).

% sorted_remove1
thf(fact_5985_sorted__remove1,axiom,
    ! [Xs: list_nat,A: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).

% sorted_remove1
thf(fact_5986_sorted__remove1,axiom,
    ! [Xs: list_int,A: int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( remove1_int @ A @ Xs ) ) ) ).

% sorted_remove1
thf(fact_5987_nth__butlast,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ ( butlast_VEBT_VEBTi @ Xs ) ) )
     => ( ( nth_VEBT_VEBTi @ ( butlast_VEBT_VEBTi @ Xs ) @ N )
        = ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_5988_nth__butlast,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ ( butlast_int @ Xs ) ) )
     => ( ( nth_int @ ( butlast_int @ Xs ) @ N )
        = ( nth_int @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_5989_nth__butlast,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ ( butlast_VEBT_VEBT @ Xs ) ) )
     => ( ( nth_VEBT_VEBT @ ( butlast_VEBT_VEBT @ Xs ) @ N )
        = ( nth_VEBT_VEBT @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_5990_nth__butlast,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ ( butlast_real @ Xs ) ) )
     => ( ( nth_real @ ( butlast_real @ Xs ) @ N )
        = ( nth_real @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_5991_nth__butlast,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ ( butlast_o @ Xs ) ) )
     => ( ( nth_o @ ( butlast_o @ Xs ) @ N )
        = ( nth_o @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_5992_nth__butlast,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
     => ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
        = ( nth_nat @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_5993_take__butlast,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( take_VEBT_VEBT @ N @ ( butlast_VEBT_VEBT @ Xs ) )
        = ( take_VEBT_VEBT @ N @ Xs ) ) ) ).

% take_butlast
thf(fact_5994_take__butlast,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( ( take_real @ N @ ( butlast_real @ Xs ) )
        = ( take_real @ N @ Xs ) ) ) ).

% take_butlast
thf(fact_5995_take__butlast,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ( take_o @ N @ ( butlast_o @ Xs ) )
        = ( take_o @ N @ Xs ) ) ) ).

% take_butlast
thf(fact_5996_take__butlast,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
        = ( take_nat @ N @ Xs ) ) ) ).

% take_butlast
thf(fact_5997_distinct__butlast__swap,axiom,
    ! [Pq: list_nat,I: nat] :
      ( ( distinct_nat @ Pq )
     => ( distinct_nat @ ( butlast_nat @ ( list_update_nat @ Pq @ I @ ( last_nat @ Pq ) ) ) ) ) ).

% distinct_butlast_swap
thf(fact_5998_distinct__butlast__swap,axiom,
    ! [Pq: list_VEBT_VEBTi,I: nat] :
      ( ( distinct_VEBT_VEBTi @ Pq )
     => ( distinct_VEBT_VEBTi @ ( butlast_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Pq @ I @ ( last_VEBT_VEBTi @ Pq ) ) ) ) ) ).

% distinct_butlast_swap
thf(fact_5999_distinct__butlast__swap,axiom,
    ! [Pq: list_VEBT_VEBT,I: nat] :
      ( ( distinct_VEBT_VEBT @ Pq )
     => ( distinct_VEBT_VEBT @ ( butlast_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Pq @ I @ ( last_VEBT_VEBT @ Pq ) ) ) ) ) ).

% distinct_butlast_swap
thf(fact_6000_butlast__take,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( butlast_VEBT_VEBT @ ( take_VEBT_VEBT @ N @ Xs ) )
        = ( take_VEBT_VEBT @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).

% butlast_take
thf(fact_6001_butlast__take,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( ( butlast_real @ ( take_real @ N @ Xs ) )
        = ( take_real @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).

% butlast_take
thf(fact_6002_butlast__take,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ( butlast_o @ ( take_o @ N @ Xs ) )
        = ( take_o @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).

% butlast_take
thf(fact_6003_butlast__take,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( butlast_nat @ ( take_nat @ N @ Xs ) )
        = ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).

% butlast_take
thf(fact_6004_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( linord2324613341767563021nteger @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) )
        = ( remove1_Code_integer @ X @ ( linord2324613341767563021nteger @ A4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_6005_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( linord2612477271533052124et_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) )
        = ( remove1_int @ X @ ( linord2612477271533052124et_int @ A4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_6006_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_o,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( linord3142498349692569832_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) )
        = ( remove1_o @ X @ ( linord3142498349692569832_set_o @ A4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_6007_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
        = ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_6008_Cons__lenlex__iff,axiom,
    ! [M: uint32,Ms: list_uint32,N: uint32,Ns: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ ( cons_uint32 @ M @ Ms ) @ ( cons_uint32 @ N @ Ns ) ) @ ( lenlex_uint32 @ R ) )
      = ( ( ord_less_nat @ ( size_s4844771616002835472uint32 @ Ms ) @ ( size_s4844771616002835472uint32 @ Ns ) )
        | ( ( ( size_s4844771616002835472uint32 @ Ms )
            = ( size_s4844771616002835472uint32 @ Ns ) )
          & ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Ms @ Ns ) @ ( lenlex_uint32 @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_6009_Cons__lenlex__iff,axiom,
    ! [M: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
        | ( ( ( size_size_list_int @ Ms )
            = ( size_size_list_int @ Ns ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_6010_Cons__lenlex__iff,axiom,
    ! [M: vEBT_VEBT,Ms: list_VEBT_VEBT,N: vEBT_VEBT,Ns: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ ( cons_VEBT_VEBT @ M @ Ms ) @ ( cons_VEBT_VEBT @ N @ Ns ) ) @ ( lenlex_VEBT_VEBT @ R ) )
      = ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ms ) @ ( size_s6755466524823107622T_VEBT @ Ns ) )
        | ( ( ( size_s6755466524823107622T_VEBT @ Ms )
            = ( size_s6755466524823107622T_VEBT @ Ns ) )
          & ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Ms @ Ns ) @ ( lenlex_VEBT_VEBT @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_6011_Cons__lenlex__iff,axiom,
    ! [M: real,Ms: list_real,N: real,Ns: list_real,R: set_Pr6218003697084177305l_real] :
      ( ( member6584958104391596930t_real @ ( produc1408950526243324945t_real @ ( cons_real @ M @ Ms ) @ ( cons_real @ N @ Ns ) ) @ ( lenlex_real @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_real @ Ms ) @ ( size_size_list_real @ Ns ) )
        | ( ( ( size_size_list_real @ Ms )
            = ( size_size_list_real @ Ns ) )
          & ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member6584958104391596930t_real @ ( produc1408950526243324945t_real @ Ms @ Ns ) @ ( lenlex_real @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_6012_Cons__lenlex__iff,axiom,
    ! [M: $o,Ms: list_o,N: $o,Ns: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ M @ Ms ) @ ( cons_o @ N @ Ns ) ) @ ( lenlex_o @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_o @ Ms ) @ ( size_size_list_o @ Ns ) )
        | ( ( ( size_size_list_o @ Ms )
            = ( size_size_list_o @ Ns ) )
          & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ms @ Ns ) @ ( lenlex_o @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_6013_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_6014_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi,R: set_Pr2227491710730465451_VEBTi] :
      ( ( member4173000155140927252_VEBTi @ ( produc4384243565435462691_VEBTi @ Xs @ Ys ) @ ( listrel1_VEBT_VEBTi @ R ) )
      = ( ? [Y4: vEBT_VEBTi,N6: nat] :
            ( ( member660371905731732212_VEBTi @ ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ N6 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N6 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
            & ( Ys
              = ( list_u6098035379799741383_VEBTi @ Xs @ N6 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_6015_listrel1__iff__update,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Xs @ Ys ) @ ( listrel1_uint32 @ R ) )
      = ( ? [Y4: uint32,N6: nat] :
            ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ ( nth_uint32 @ Xs @ N6 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N6 @ ( size_s4844771616002835472uint32 @ Xs ) )
            & ( Ys
              = ( list_update_uint32 @ Xs @ N6 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_6016_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
      = ( ? [Y4: int,N6: nat] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N6 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N6 @ ( size_size_list_int @ Xs ) )
            & ( Ys
              = ( list_update_int @ Xs @ N6 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_6017_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R ) )
      = ( ? [Y4: vEBT_VEBT,N6: nat] :
            ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N6 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N6 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( Ys
              = ( list_u1324408373059187874T_VEBT @ Xs @ N6 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_6018_listrel1__iff__update,axiom,
    ! [Xs: list_real,Ys: list_real,R: set_Pr6218003697084177305l_real] :
      ( ( member6584958104391596930t_real @ ( produc1408950526243324945t_real @ Xs @ Ys ) @ ( listrel1_real @ R ) )
      = ( ? [Y4: real,N6: nat] :
            ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ ( nth_real @ Xs @ N6 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N6 @ ( size_size_list_real @ Xs ) )
            & ( Ys
              = ( list_update_real @ Xs @ N6 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_6019_listrel1__iff__update,axiom,
    ! [Xs: list_o,Ys: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R ) )
      = ( ? [Y4: $o,N6: nat] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N6 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N6 @ ( size_size_list_o @ Xs ) )
            & ( Ys
              = ( list_update_o @ Xs @ N6 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_6020_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
      = ( ? [Y4: nat,N6: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N6 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N6 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N6 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_6021_total__on__singleton,axiom,
    ! [X: vEBT_VEBT] : ( total_on_VEBT_VEBT @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) @ ( insert494605675473494903T_VEBT @ ( produc537772716801021591T_VEBT @ X @ X ) @ bot_bo5088076668136028147T_VEBT ) ) ).

% total_on_singleton
thf(fact_6022_total__on__singleton,axiom,
    ! [X: uint32] : ( total_on_uint32 @ ( insert_uint32 @ X @ bot_bot_set_uint32 ) @ ( insert4454361187789264009uint32 @ ( produc1400373151660368625uint32 @ X @ X ) @ bot_bo8438649754162204037uint32 ) ) ).

% total_on_singleton
thf(fact_6023_total__on__singleton,axiom,
    ! [X: nat] : ( total_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).

% total_on_singleton
thf(fact_6024_total__on__singleton,axiom,
    ! [X: int] : ( total_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).

% total_on_singleton
thf(fact_6025_total__on__singleton,axiom,
    ! [X: $o] : ( total_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).

% total_on_singleton
thf(fact_6026_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Ys ) )
       => ( ( nth_Pr6329974346453275474_VEBTi @ ( zip_VE793581609497812771_VEBTi @ Xs @ Ys ) @ I )
          = ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6027_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr2733638074192486473Ti_int @ ( zip_VEBT_VEBTi_int @ Xs @ Ys ) @ I )
          = ( produc7190175284220460154Ti_int @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6028_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Ys ) )
       => ( ( nth_Pr5092243222550848201_VEBTi @ ( zip_int_VEBT_VEBTi @ Xs @ Ys ) @ I )
          = ( produc4497823428076633978_VEBTi @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6029_nth__zip,axiom,
    ! [I: nat,Xs: list_uint32,Ys: list_uint32] :
      ( ( ord_less_nat @ I @ ( size_s4844771616002835472uint32 @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s4844771616002835472uint32 @ Ys ) )
       => ( ( nth_Pr6139322253122866368uint32 @ ( zip_uint32_uint32 @ Xs @ Ys ) @ I )
          = ( produc1400373151660368625uint32 @ ( nth_uint32 @ Xs @ I ) @ ( nth_uint32 @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6030_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys ) @ I )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6031_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr8725177398587324397T_VEBT @ ( zip_VE7413257051550508102T_VEBT @ Xs @ Ys ) @ I )
          = ( produc7053807326796202854T_VEBT @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6032_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) @ I )
          = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6033_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_real] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Ys ) )
       => ( ( nth_Pr3433448822664029129i_real @ ( zip_VEBT_VEBTi_real @ Xs @ Ys ) @ I )
          = ( produc8457151488442208762i_real @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_real @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6034_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_real] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Ys ) )
       => ( ( nth_Pr731366597535767232t_real @ ( zip_int_real @ Xs @ Ys ) @ I )
          = ( produc801115645435158769t_real @ ( nth_int @ Xs @ I ) @ ( nth_real @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6035_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Pr3306050735993963089EBTi_o @ ( zip_VEBT_VEBTi_o @ Xs @ Ys ) @ I )
          = ( produc8194178580519725514EBTi_o @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_6036_rule__at__index,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L7265847600308530106T_VEBT @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L7265847600308530106T_VEBT @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6037_rule__at__index,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > nat > assn,Xs: list_VEBT_VEBTi,Xsi: list_nat,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L8930081998596925642Ti_nat @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L8930081998596925642Ti_nat @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6038_rule__at__index,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L1891944875198410415_VEBTi @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L1891944875198410415_VEBTi @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6039_rule__at__index,axiom,
    ! [P2: assn,A4: vEBT_VEBTi > int > assn,Xs: list_VEBT_VEBTi,Xsi: list_int,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L8927591528087875366Ti_int @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L8927591528087875366Ti_int @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6040_rule__at__index,axiom,
    ! [P2: assn,A4: int > vEBT_VEBT > assn,Xs: list_int,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L1664421287176695555T_VEBT @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2018189785592951398T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2018189785592951398T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L1664421287176695555T_VEBT @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6041_rule__at__index,axiom,
    ! [P2: assn,A4: int > nat > assn,Xs: list_int,Xsi: list_nat,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L77084186935402305nt_nat @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L77084186935402305nt_nat @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6042_rule__at__index,axiom,
    ! [P2: assn,A4: int > vEBT_VEBTi > assn,Xs: list_int,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L6235239671944049190_VEBTi @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L114188773329725699_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L114188773329725699_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L6235239671944049190_VEBTi @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6043_rule__at__index,axiom,
    ! [P2: assn,A4: int > int > assn,Xs: list_int,Xsi: list_int,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L74593716426352029nt_int @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L74593716426352029nt_int @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6044_rule__at__index,axiom,
    ! [P2: assn,A4: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6045_rule__at__index,axiom,
    ! [P2: assn,A4: vEBT_VEBT > nat > assn,Xs: list_VEBT_VEBT,Xsi: list_nat,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: vEBT_VEBTi > assn,F4: vEBT_VEBTi > assn] :
      ( ( entails @ P2 @ ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A4 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ F2 ) @ C2 @ Q4 )
         => ( ! [R4: vEBT_VEBTi] : ( entails @ ( Q4 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A4 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A4 @ Xs @ Xsi ) ) @ ( F4 @ R4 ) ) )
           => ( hoare_1429296392585015714_VEBTi @ P2 @ C2
              @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A4 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_6046_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_6047_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_6048_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_6049_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_6050_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_6051_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_6052_image__ident,axiom,
    ! [Y7: set_nat] :
      ( ( image_nat_nat
        @ ^ [X4: nat] : X4
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_6053_image__ident,axiom,
    ! [Y7: set_int] :
      ( ( image_int_int
        @ ^ [X4: int] : X4
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_6054_finite__Collect__disjI,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6055_finite__Collect__disjI,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6056_finite__Collect__disjI,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        & ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6057_finite__Collect__disjI,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P2 ) )
        & ( finite_finite_int @ ( collect_int @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6058_finite__Collect__disjI,axiom,
    ! [P2: complex > $o,Q2: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6059_finite__Collect__disjI,axiom,
    ! [P2: code_integer > $o,Q2: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P2 ) )
        & ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6060_finite__Collect__conjI,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q2 ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6061_finite__Collect__conjI,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q2 ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6062_finite__Collect__conjI,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        | ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6063_finite__Collect__conjI,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P2 ) )
        | ( finite_finite_int @ ( collect_int @ Q2 ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6064_finite__Collect__conjI,axiom,
    ! [P2: complex > $o,Q2: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q2 ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6065_finite__Collect__conjI,axiom,
    ! [P2: code_integer > $o,Q2: code_integer > $o] :
      ( ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P2 ) )
        | ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q2 ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6066_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y4: nat] :
                ( ( vEBT_vebt_member @ T @ Y4 )
                & ( ord_less_nat @ X @ Y4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_6067_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y4: nat] :
                ( ( vEBT_vebt_member @ T @ Y4 )
                & ( ord_less_nat @ Y4 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_6068_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C2: nat,D: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( ( ord_less_eq_nat @ C2 @ D )
         => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ C2 ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ) ) ) ).

% foldr_mono
thf(fact_6069_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6070_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6071_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_6072_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6073_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_6074_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_6075_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_VEBT_VEBT,A2: list_VEBT_VEBT,B2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_s6755466524823107622T_VEBT @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_s6755466524823107622T_VEBT @ B2 ) )
       => ( ( ( zip_VE537291747668921783T_VEBT @ A @ B )
            = ( zip_VE537291747668921783T_VEBT @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6076_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_real,A2: list_VEBT_VEBT,B2: list_real] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_size_list_real @ B2 ) )
       => ( ( ( zip_VEBT_VEBT_real @ A @ B )
            = ( zip_VEBT_VEBT_real @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6077_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_o,A2: list_VEBT_VEBT,B2: list_o] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_VEBT_VEBT_o @ A @ B )
            = ( zip_VEBT_VEBT_o @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6078_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_nat,A2: list_VEBT_VEBT,B2: list_nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_VEBT_VEBT_nat @ A @ B )
            = ( zip_VEBT_VEBT_nat @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6079_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_VEBT_VEBT,A2: list_real,B2: list_VEBT_VEBT] :
      ( ( ( size_size_list_real @ A )
        = ( size_s6755466524823107622T_VEBT @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_s6755466524823107622T_VEBT @ B2 ) )
       => ( ( ( zip_real_VEBT_VEBT @ A @ B )
            = ( zip_real_VEBT_VEBT @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6080_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_real,A2: list_real,B2: list_real] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_size_list_real @ B2 ) )
       => ( ( ( zip_real_real @ A @ B )
            = ( zip_real_real @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6081_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_o,A2: list_real,B2: list_o] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_real_o @ A @ B )
            = ( zip_real_o @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6082_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_nat,A2: list_real,B2: list_nat] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_real_nat @ A @ B )
            = ( zip_real_nat @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6083_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_VEBT_VEBT,A2: list_o,B2: list_VEBT_VEBT] :
      ( ( ( size_size_list_o @ A )
        = ( size_s6755466524823107622T_VEBT @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_s6755466524823107622T_VEBT @ B2 ) )
       => ( ( ( zip_o_VEBT_VEBT @ A @ B )
            = ( zip_o_VEBT_VEBT @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6084_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_real,A2: list_o,B2: list_real] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_real @ B2 ) )
       => ( ( ( zip_o_real @ A @ B )
            = ( zip_o_real @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_6085_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_6086_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ D ) @ D ) ) ) ).

% foldr_zero
thf(fact_6087_sum_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A4 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_6088_sum_Oneutral__const,axiom,
    ! [A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A4 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_6089_sum_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A4 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_6090_sum_Oneutral__const,axiom,
    ! [A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A4 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_6091_singleton__conv2,axiom,
    ! [A: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ( ^ [Y6: vEBT_VEBT,Z4: vEBT_VEBT] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv2
thf(fact_6092_singleton__conv2,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ( ^ [Y6: complex,Z4: complex] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv2
thf(fact_6093_singleton__conv2,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_6094_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_6095_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_6096_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_6097_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_6098_singleton__conv,axiom,
    ! [A: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] : ( X4 = A ) )
      = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv
thf(fact_6099_singleton__conv,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ^ [X4: complex] : ( X4 = A ) )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_6100_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( X4 = A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_6101_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( X4 = A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_6102_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( X4 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_6103_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X4: int] : ( X4 = A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_6104_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X4: $o] : ( X4 = A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_6105_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N6: nat] : ( ord_less_nat @ N6 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_6106_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N6: nat] : ( ord_less_eq_nat @ N6 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_6107_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_6108_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_6109_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_6110_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_6111_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_6112_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6113_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6114_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6115_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6116_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6117_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6118_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri2565882477558803405uint32 @ zero_zero_nat )
    = zero_zero_uint32 ) ).

% semiring_1_class.of_nat_0
thf(fact_6119_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_6120_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_6121_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_6122_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_6123_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% semiring_1_class.of_nat_0
thf(fact_6124_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_6125_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_6126_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_6127_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_6128_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_6129_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_6130_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_6131_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_6132_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_6133_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_6134_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_6135_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_6136_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_6137_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_6138_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_add
thf(fact_6139_of__nat__1,axiom,
    ( ( semiri2565882477558803405uint32 @ one_one_nat )
    = one_one_uint32 ) ).

% of_nat_1
thf(fact_6140_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_6141_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_6142_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_6143_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_6144_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_6145_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6146_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6147_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6148_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6149_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6150_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6151_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6152_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6153_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6154_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri4939895301339042750nteger @ N )
        = one_one_Code_integer )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6155_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri2565882477558803405uint32 @ ( times_times_nat @ M @ N ) )
      = ( times_times_uint32 @ ( semiri2565882477558803405uint32 @ M ) @ ( semiri2565882477558803405uint32 @ N ) ) ) ).

% of_nat_mult
thf(fact_6156_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_6157_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_6158_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_6159_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mult
thf(fact_6160_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6161_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6162_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6163_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6164_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6165_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_6166_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K3: vEBT_VEBT] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K3: vEBT_VEBT] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_6167_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > uint32] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K3: real] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K3: real] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_6168_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > uint32] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups833757482993574392uint32
              @ ^ [K3: nat] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups833757482993574392uint32
              @ ^ [K3: nat] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_6169_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > uint32] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K3: int] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K3: int] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_6170_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > uint32] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K3: complex] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K3: complex] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_6171_sum_Odelta_H,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K3: code_integer] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K3: code_integer] : ( if_uint32 @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta'
thf(fact_6172_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6173_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6174_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6175_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6176_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K3: vEBT_VEBT] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8325533452322294502uint32
              @ ^ [K3: vEBT_VEBT] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_6177_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > uint32] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K3: real] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5944083974425963860uint32
              @ ^ [K3: real] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_6178_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > uint32] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups833757482993574392uint32
              @ ^ [K3: nat] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups833757482993574392uint32
              @ ^ [K3: nat] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_6179_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > uint32] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K3: int] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups5712668689793887828uint32
              @ ^ [K3: int] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_6180_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > uint32] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K3: complex] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups8736914816313324502uint32
              @ ^ [K3: complex] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_6181_sum_Odelta,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K3: code_integer] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups8847630953604152069uint32
              @ ^ [K3: code_integer] : ( if_uint32 @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_uint32 )
              @ S3 )
            = zero_zero_uint32 ) ) ) ) ).

% sum.delta
thf(fact_6182_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6183_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6184_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6185_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6186_foldl__length,axiom,
    ! [L: list_VEBT_VEBT] :
      ( ( foldl_nat_VEBT_VEBT
        @ ^ [I3: nat,X4: vEBT_VEBT] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_s6755466524823107622T_VEBT @ L ) ) ).

% foldl_length
thf(fact_6187_foldl__length,axiom,
    ! [L: list_real] :
      ( ( foldl_nat_real
        @ ^ [I3: nat,X4: real] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_size_list_real @ L ) ) ).

% foldl_length
thf(fact_6188_foldl__length,axiom,
    ! [L: list_o] :
      ( ( foldl_nat_o
        @ ^ [I3: nat,X4: $o] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_size_list_o @ L ) ) ).

% foldl_length
thf(fact_6189_foldl__length,axiom,
    ! [L: list_nat] :
      ( ( foldl_nat_nat
        @ ^ [I3: nat,X4: nat] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_size_list_nat @ L ) ) ).

% foldl_length
thf(fact_6190_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_6191_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6192_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6193_finite__Collect__subsets,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B6: set_Code_integer] : ( ord_le7084787975880047091nteger @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6194_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6195_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_6196_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_6197_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_6198_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_6199_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_6200_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri2565882477558803405uint32 @ ( suc @ M ) )
      = ( plus_plus_uint32 @ one_one_uint32 @ ( semiri2565882477558803405uint32 @ M ) ) ) ).

% of_nat_Suc
thf(fact_6201_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_6202_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_6203_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_6204_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_6205_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_6206_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6207_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6208_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6209_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6210_single__Diff__lessThan,axiom,
    ! [K: $o] :
      ( ( minus_minus_set_o @ ( insert_o @ K @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K ) )
      = ( insert_o @ K @ bot_bot_set_o ) ) ).

% single_Diff_lessThan
thf(fact_6211_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_6212_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_6213_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_6214_Cons__listrel1__Cons,axiom,
    ! [X: $o,Xs: list_o,Y: $o,Ys: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs ) @ ( cons_o @ Y @ Ys ) ) @ ( listrel1_o @ R ) )
      = ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_6215_Cons__listrel1__Cons,axiom,
    ! [X: uint32,Xs: list_uint32,Y: uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ ( cons_uint32 @ X @ Xs ) @ ( cons_uint32 @ Y @ Ys ) ) @ ( listrel1_uint32 @ R ) )
      = ( ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Xs @ Ys ) @ ( listrel1_uint32 @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_6216_Cons__listrel1__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_6217_Cons__listrel1__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_6218_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( zip_int_nat @ ( cons_int @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P7512249878480867347nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ ( zip_int_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6219_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: $o,Ys: list_o] :
      ( ( zip_int_o @ ( cons_int @ X @ Xs ) @ ( cons_o @ Y @ Ys ) )
      = ( cons_P7321330006258091179_int_o @ ( product_Pair_int_o @ X @ Y ) @ ( zip_int_o @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6220_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( zip_nat_int @ ( cons_nat @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ X @ Y ) @ ( zip_nat_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6221_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: $o,Ys: list_o] :
      ( ( zip_nat_o @ ( cons_nat @ X @ Xs ) @ ( cons_o @ Y @ Ys ) )
      = ( cons_P9142372351690779143_nat_o @ ( product_Pair_nat_o @ X @ Y ) @ ( zip_nat_o @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6222_zip__Cons__Cons,axiom,
    ! [X: $o,Xs: list_o,Y: int,Ys: list_int] :
      ( ( zip_o_int @ ( cons_o @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P1455986808126089405_o_int @ ( product_Pair_o_int @ X @ Y ) @ ( zip_o_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6223_zip__Cons__Cons,axiom,
    ! [X: $o,Xs: list_o,Y: nat,Ys: list_nat] :
      ( ( zip_o_nat @ ( cons_o @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P5633837827635286113_o_nat @ ( product_Pair_o_nat @ X @ Y ) @ ( zip_o_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6224_zip__Cons__Cons,axiom,
    ! [X: $o,Xs: list_o,Y: $o,Ys: list_o] :
      ( ( zip_o_o @ ( cons_o @ X @ Xs ) @ ( cons_o @ Y @ Ys ) )
      = ( cons_P8766293264717362397od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( zip_o_o @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6225_zip__Cons__Cons,axiom,
    ! [X: uint32,Xs: list_uint32,Y: uint32,Ys: list_uint32] :
      ( ( zip_uint32_uint32 @ ( cons_uint32 @ X @ Xs ) @ ( cons_uint32 @ Y @ Ys ) )
      = ( cons_P3149448846263281007uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ ( zip_uint32_uint32 @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6226_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6227_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( zip_int_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_6228_Max__const,axiom,
    ! [A4: set_nat,C2: int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_nat_int
              @ ^ [Uu3: nat] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_6229_Max__const,axiom,
    ! [A4: set_int,C2: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_int_int
              @ ^ [Uu3: int] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_6230_Max__const,axiom,
    ! [A4: set_complex,C2: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_complex_nat
              @ ^ [Uu3: complex] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_6231_Max__const,axiom,
    ! [A4: set_Code_integer,C2: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_951025933927791156er_nat
              @ ^ [Uu3: code_integer] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_6232_Max__const,axiom,
    ! [A4: set_nat,C2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_nat_nat
              @ ^ [Uu3: nat] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_6233_Max__const,axiom,
    ! [A4: set_int,C2: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_int_nat
              @ ^ [Uu3: int] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_6234_Max__const,axiom,
    ! [A4: set_o,C2: nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_o_nat
              @ ^ [Uu3: $o] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_6235_Min__const,axiom,
    ! [A4: set_nat,C2: int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( lattic8718645017227715691in_int
            @ ( image_nat_int
              @ ^ [Uu3: nat] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6236_Min__const,axiom,
    ! [A4: set_int,C2: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( lattic8718645017227715691in_int
            @ ( image_int_int
              @ ^ [Uu3: int] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6237_Min__const,axiom,
    ! [A4: set_complex,C2: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_complex_nat
              @ ^ [Uu3: complex] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6238_Min__const,axiom,
    ! [A4: set_Code_integer,C2: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_951025933927791156er_nat
              @ ^ [Uu3: code_integer] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6239_Min__const,axiom,
    ! [A4: set_nat,C2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_nat_nat
              @ ^ [Uu3: nat] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6240_Min__const,axiom,
    ! [A4: set_int,C2: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_int_nat
              @ ^ [Uu3: int] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6241_Min__const,axiom,
    ! [A4: set_o,C2: nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_o_nat
              @ ^ [Uu3: $o] : C2
              @ A4 ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6242_foldr__length,axiom,
    ! [L: list_VEBT_VEBT] :
      ( ( foldr_VEBT_VEBT_nat
        @ ^ [X4: vEBT_VEBT] : suc
        @ L
        @ zero_zero_nat )
      = ( size_s6755466524823107622T_VEBT @ L ) ) ).

% foldr_length
thf(fact_6243_foldr__length,axiom,
    ! [L: list_real] :
      ( ( foldr_real_nat
        @ ^ [X4: real] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_real @ L ) ) ).

% foldr_length
thf(fact_6244_foldr__length,axiom,
    ! [L: list_o] :
      ( ( foldr_o_nat
        @ ^ [X4: $o] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_o @ L ) ) ).

% foldr_length
thf(fact_6245_foldr__length,axiom,
    ! [L: list_nat] :
      ( ( foldr_nat_nat
        @ ^ [X4: nat] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_nat @ L ) ) ).

% foldr_length
thf(fact_6246_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6247_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6248_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6249_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6250_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6251_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6252_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6253_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6254_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6255_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6256_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6257_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6258_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6259_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6260_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6261_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6262_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6263_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6264_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6265_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6266_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6267_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6268_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6269_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6270_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6271_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6272_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6273_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6274_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6275_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6276_sum__zero__power,axiom,
    ! [A4: set_nat,C2: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A4 )
          = ( C2 @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_6277_sum__zero__power,axiom,
    ! [A4: set_nat,C2: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A4 )
          = ( C2 @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_6278_sum__zero__power,axiom,
    ! [A4: set_nat,C2: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A4 )
          = ( C2 @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_6279_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ( size_s3451745648224563538omplex @ Xs2 )
                = N )
              & ( distinct_complex @ Xs2 )
              & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6280_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_Code_integer,N: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs2: list_Code_integer] :
              ( ( ( size_s3445333598471063425nteger @ Xs2 )
                = N )
              & ( distin1543349897113766820nteger @ Xs2 )
              & ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs2 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6281_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N )
              & ( distinct_VEBT_VEBT @ Xs2 )
              & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6282_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_real,N: nat] :
      ( ( finite_finite_real @ A4 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs2: list_real] :
              ( ( ( size_size_list_real @ Xs2 )
                = N )
              & ( distinct_real @ Xs2 )
              & ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6283_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ( size_size_list_o @ Xs2 )
                = N )
              & ( distinct_o @ Xs2 )
              & ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6284_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = N )
              & ( distinct_nat @ Xs2 )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6285_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ( size_size_list_int @ Xs2 )
                = N )
              & ( distinct_int @ Xs2 )
              & ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6286_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_member @ T @ X ) ) ) ) ) ).

% vebt_memberi'_rf_abstr
thf(fact_6287_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_succ @ T @ X ) ) ) ) ) ) ).

% vebt_succi'_rf_abstr
thf(fact_6288_vebt__pred_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_pred @ T @ X ) ) ) ) ) ) ).

% vebt_pred'_rf_abstr
thf(fact_6289_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) ) ).

% vebt_inserti'_rf_abstr
thf(fact_6290_pred__equals__eq2,axiom,
    ! [R2: set_Pr1773385645901665561uint32,S3: set_Pr1773385645901665561uint32] :
      ( ( ( ^ [X4: uint32,Y4: uint32] : ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: uint32,Y4: uint32] : ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_6291_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_6292_pred__equals__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_6293_pred__equals__eq2,axiom,
    ! [R2: set_Pr3286484037609594932et_nat,S3: set_Pr3286484037609594932et_nat] :
      ( ( ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_6294_pred__equals__eq2,axiom,
    ! [R2: set_Pr8536935166611901872et_nat,S3: set_Pr8536935166611901872et_nat] :
      ( ( ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_6295_pred__subset__eq,axiom,
    ! [R2: set_nat,S3: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
      = ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_6296_pred__subset__eq,axiom,
    ! [R2: set_VEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( ord_le418104280809901481VEBT_o
        @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ R2 )
        @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ S3 ) )
      = ( ord_le4337996190870823476T_VEBT @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_6297_pred__subset__eq,axiom,
    ! [R2: set_real,S3: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X4: real] : ( member_real @ X4 @ R2 )
        @ ^ [X4: real] : ( member_real @ X4 @ S3 ) )
      = ( ord_less_eq_set_real @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_6298_pred__subset__eq,axiom,
    ! [R2: set_set_nat,S3: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R2 )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S3 ) )
      = ( ord_le6893508408891458716et_nat @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_6299_pred__subset__eq,axiom,
    ! [R2: set_int,S3: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R2 )
        @ ^ [X4: int] : ( member_int @ X4 @ S3 ) )
      = ( ord_less_eq_set_int @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_6300_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6301_less__eq__set__def,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( ord_le418104280809901481VEBT_o
          @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ A6 )
          @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6302_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A6 )
          @ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6303_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6304_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A6 )
          @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6305_Collect__subset,axiom,
    ! [A4: set_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ord_le4337996190870823476T_VEBT
      @ ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_6306_Collect__subset,axiom,
    ! [A4: set_real,P2: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_6307_Collect__subset,axiom,
    ! [A4: set_complex,P2: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X4: complex] :
            ( ( member_complex @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_6308_Collect__subset,axiom,
    ! [A4: set_list_nat,P2: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_6309_Collect__subset,axiom,
    ! [A4: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_6310_Collect__subset,axiom,
    ! [A4: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_6311_Collect__subset,axiom,
    ! [A4: set_int,P2: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_6312_prop__restrict,axiom,
    ! [X: vEBT_VEBT,Z6: set_VEBT_VEBT,X6: set_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ X @ Z6 )
     => ( ( ord_le4337996190870823476T_VEBT @ Z6
          @ ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ X6 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_6313_prop__restrict,axiom,
    ! [X: real,Z6: set_real,X6: set_real,P2: real > $o] :
      ( ( member_real @ X @ Z6 )
     => ( ( ord_less_eq_set_real @ Z6
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ X6 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_6314_prop__restrict,axiom,
    ! [X: complex,Z6: set_complex,X6: set_complex,P2: complex > $o] :
      ( ( member_complex @ X @ Z6 )
     => ( ( ord_le211207098394363844omplex @ Z6
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ X6 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_6315_prop__restrict,axiom,
    ! [X: list_nat,Z6: set_list_nat,X6: set_list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ X @ Z6 )
     => ( ( ord_le6045566169113846134st_nat @ Z6
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ X6 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_6316_prop__restrict,axiom,
    ! [X: set_nat,Z6: set_set_nat,X6: set_set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ X @ Z6 )
     => ( ( ord_le6893508408891458716et_nat @ Z6
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ X6 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_6317_prop__restrict,axiom,
    ! [X: nat,Z6: set_nat,X6: set_nat,P2: nat > $o] :
      ( ( member_nat @ X @ Z6 )
     => ( ( ord_less_eq_set_nat @ Z6
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ X6 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_6318_prop__restrict,axiom,
    ! [X: int,Z6: set_int,X6: set_int,P2: int > $o] :
      ( ( member_int @ X @ Z6 )
     => ( ( ord_less_eq_set_int @ Z6
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ X6 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_6319_Collect__restrict,axiom,
    ! [X6: set_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ord_le4337996190870823476T_VEBT
      @ ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ X6 )
            & ( P2 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_6320_Collect__restrict,axiom,
    ! [X6: set_real,P2: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ X6 )
            & ( P2 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_6321_Collect__restrict,axiom,
    ! [X6: set_complex,P2: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X4: complex] :
            ( ( member_complex @ X4 @ X6 )
            & ( P2 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_6322_Collect__restrict,axiom,
    ! [X6: set_list_nat,P2: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ X6 )
            & ( P2 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_6323_Collect__restrict,axiom,
    ! [X6: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ X6 )
            & ( P2 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_6324_Collect__restrict,axiom,
    ! [X6: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ X6 )
            & ( P2 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_6325_Collect__restrict,axiom,
    ! [X6: set_int,P2: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ X6 )
            & ( P2 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_6326_subset__CollectI,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,Q2: vEBT_VEBT > $o,P2: vEBT_VEBT > $o] :
      ( ( ord_le4337996190870823476T_VEBT @ B4 @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le4337996190870823476T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_6327_subset__CollectI,axiom,
    ! [B4: set_real,A4: set_real,Q2: real > $o,P2: real > $o] :
      ( ( ord_less_eq_set_real @ B4 @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_6328_subset__CollectI,axiom,
    ! [B4: set_complex,A4: set_complex,Q2: complex > $o,P2: complex > $o] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le211207098394363844omplex
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_6329_subset__CollectI,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,Q2: list_nat > $o,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B4 @ A4 )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le6045566169113846134st_nat
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_6330_subset__CollectI,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,Q2: set_nat > $o,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_6331_subset__CollectI,axiom,
    ! [B4: set_nat,A4: set_nat,Q2: nat > $o,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_6332_subset__CollectI,axiom,
    ! [B4: set_int,A4: set_int,Q2: int > $o,P2: int > $o] :
      ( ( ord_less_eq_set_int @ B4 @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_6333_subset__Collect__iff,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ord_le4337996190870823476T_VEBT @ B4 @ A4 )
     => ( ( ord_le4337996190870823476T_VEBT @ B4
          @ ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_6334_subset__Collect__iff,axiom,
    ! [B4: set_real,A4: set_real,P2: real > $o] :
      ( ( ord_less_eq_set_real @ B4 @ A4 )
     => ( ( ord_less_eq_set_real @ B4
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: real] :
              ( ( member_real @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_6335_subset__Collect__iff,axiom,
    ! [B4: set_complex,A4: set_complex,P2: complex > $o] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_6336_subset__Collect__iff,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B4 @ A4 )
     => ( ( ord_le6045566169113846134st_nat @ B4
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: list_nat] :
              ( ( member_list_nat @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_6337_subset__Collect__iff,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
     => ( ( ord_le6893508408891458716et_nat @ B4
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: set_nat] :
              ( ( member_set_nat @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_6338_subset__Collect__iff,axiom,
    ! [B4: set_nat,A4: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( ord_less_eq_set_nat @ B4
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_6339_subset__Collect__iff,axiom,
    ! [B4: set_int,A4: set_int,P2: int > $o] :
      ( ( ord_less_eq_set_int @ B4 @ A4 )
     => ( ( ord_less_eq_set_int @ B4
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_6340_insert__compr,axiom,
    ( insert_o
    = ( ^ [A7: $o,B6: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( X4 = A7 )
              | ( member_o @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6341_insert__compr,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A7: vEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X4: vEBT_VEBT] :
              ( ( X4 = A7 )
              | ( member_VEBT_VEBT @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6342_insert__compr,axiom,
    ( insert_real
    = ( ^ [A7: real,B6: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( X4 = A7 )
              | ( member_real @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6343_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A7: complex,B6: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ( ( X4 = A7 )
              | ( member_complex @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6344_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A7: list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( X4 = A7 )
              | ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6345_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A7: set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( X4 = A7 )
              | ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6346_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A7: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( X4 = A7 )
              | ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6347_insert__compr,axiom,
    ( insert_int
    = ( ^ [A7: int,B6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( X4 = A7 )
              | ( member_int @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6348_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( insert_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P2 ) )
      = ( collect_VEBT_VEBT
        @ ^ [U2: vEBT_VEBT] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6349_insert__Collect,axiom,
    ! [A: $o,P2: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P2 ) )
      = ( collect_o
        @ ^ [U2: $o] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6350_insert__Collect,axiom,
    ! [A: complex,P2: complex > $o] :
      ( ( insert_complex @ A @ ( collect_complex @ P2 ) )
      = ( collect_complex
        @ ^ [U2: complex] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6351_insert__Collect,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( insert_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( collect_list_nat
        @ ^ [U2: list_nat] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6352_insert__Collect,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( insert_set_nat @ A @ ( collect_set_nat @ P2 ) )
      = ( collect_set_nat
        @ ^ [U2: set_nat] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6353_insert__Collect,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P2 ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6354_insert__Collect,axiom,
    ! [A: int,P2: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P2 ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6355_Compl__eq,axiom,
    ( uminus8041839845116263051T_VEBT
    = ( ^ [A6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X4: vEBT_VEBT] :
              ~ ( member_VEBT_VEBT @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6356_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ~ ( member_real @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6357_Compl__eq,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A6: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ~ ( member_complex @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6358_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ~ ( member_list_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6359_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ~ ( member_set_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6360_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ~ ( member_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6361_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ~ ( member_int @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6362_Collect__neg__eq,axiom,
    ! [P2: complex > $o] :
      ( ( collect_complex
        @ ^ [X4: complex] :
            ~ ( P2 @ X4 ) )
      = ( uminus8566677241136511917omplex @ ( collect_complex @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_6363_Collect__neg__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ~ ( P2 @ X4 ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_6364_Collect__neg__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ~ ( P2 @ X4 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_6365_Collect__neg__eq,axiom,
    ! [P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ~ ( P2 @ X4 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_6366_Collect__neg__eq,axiom,
    ! [P2: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ~ ( P2 @ X4 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_6367_uminus__set__def,axiom,
    ( uminus8041839845116263051T_VEBT
    = ( ^ [A6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( uminus2746543603091002386VEBT_o
            @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6368_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6369_uminus__set__def,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A6: set_complex] :
          ( collect_complex
          @ ( uminus1680532995456772888plex_o
            @ ^ [X4: complex] : ( member_complex @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6370_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6371_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6372_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6373_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6374_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_set_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6375_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_rat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6376_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_num @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6377_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6378_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_int @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6379_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X4: real] : ( ord_less_real @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6380_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6381_less__set__def,axiom,
    ( ord_le3480810397992357184T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( ord_less_VEBT_VEBT_o
          @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ A6 )
          @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6382_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A6 )
          @ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6383_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A6 )
          @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6384_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6385_Set_Oempty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X4: complex] : $false ) ) ).

% Set.empty_def
thf(fact_6386_Set_Oempty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X4: list_nat] : $false ) ) ).

% Set.empty_def
thf(fact_6387_Set_Oempty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X4: set_nat] : $false ) ) ).

% Set.empty_def
thf(fact_6388_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% Set.empty_def
thf(fact_6389_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X4: int] : $false ) ) ).

% Set.empty_def
thf(fact_6390_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X4: $o] : $false ) ) ).

% Set.empty_def
thf(fact_6391_minus__set__def,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( minus_2794559001203777698VEBT_o
            @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ A6 )
            @ ^ [X4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6392_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A6 )
            @ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6393_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X4: complex] : ( member_complex @ X4 @ A6 )
            @ ^ [X4: complex] : ( member_complex @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6394_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6395_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6396_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A6 )
            @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6397_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6398_set__diff__eq,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ A6 )
              & ~ ( member_VEBT_VEBT @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6399_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A6 )
              & ~ ( member_real @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6400_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ( ( member_complex @ X4 @ A6 )
              & ~ ( member_complex @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6401_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A6 )
              & ~ ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6402_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A6 )
              & ~ ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6403_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A6 )
              & ~ ( member_int @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6404_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
              & ~ ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6405_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_nat,R2: vEBT_VEBT > nat > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A7: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6406_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_nat,R2: real > nat > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A7: real] :
                        ( ( member_real @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6407_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_int,R2: vEBT_VEBT > int > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A7: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6408_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_int,R2: real > int > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A7: real] :
                        ( ( member_real @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6409_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_complex,R2: vEBT_VEBT > complex > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A7: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6410_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_complex,R2: real > complex > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A7: real] :
                        ( ( member_real @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6411_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_Code_integer,R2: vEBT_VEBT > code_integer > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa: code_integer] :
                  ( ( member_Code_integer @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A7: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6412_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_Code_integer,R2: real > code_integer > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa: code_integer] :
                  ( ( member_Code_integer @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A7: real] :
                        ( ( member_real @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6413_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A7: nat] :
                        ( ( member_nat @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6414_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_int,R2: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B4 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A7: nat] :
                        ( ( member_nat @ A7 @ A4 )
                        & ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6415_not__finite__existsD,axiom,
    ! [P2: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
     => ? [X_1: list_nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6416_not__finite__existsD,axiom,
    ! [P2: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
     => ? [X_1: set_nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6417_not__finite__existsD,axiom,
    ! [P2: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ? [X_1: nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6418_not__finite__existsD,axiom,
    ! [P2: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P2 ) )
     => ? [X_1: int] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6419_not__finite__existsD,axiom,
    ! [P2: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
     => ? [X_1: complex] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6420_not__finite__existsD,axiom,
    ! [P2: code_integer > $o] :
      ( ~ ( finite6017078050557962740nteger @ ( collect_Code_integer @ P2 ) )
     => ? [X_1: code_integer] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6421_imageE,axiom,
    ! [B: nat,F: nat > nat,A4: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6422_imageE,axiom,
    ! [B: nat,F: vEBT_VEBT > nat,A4: set_VEBT_VEBT] :
      ( ( member_nat @ B @ ( image_VEBT_VEBT_nat @ F @ A4 ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_VEBT_VEBT @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6423_imageE,axiom,
    ! [B: nat,F: real > nat,A4: set_real] :
      ( ( member_nat @ B @ ( image_real_nat @ F @ A4 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6424_imageE,axiom,
    ! [B: nat,F: int > nat,A4: set_int] :
      ( ( member_nat @ B @ ( image_int_nat @ F @ A4 ) )
     => ~ ! [X3: int] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_int @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6425_imageE,axiom,
    ! [B: vEBT_VEBT,F: nat > vEBT_VEBT,A4: set_nat] :
      ( ( member_VEBT_VEBT @ B @ ( image_nat_VEBT_VEBT @ F @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6426_imageE,axiom,
    ! [B: vEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B @ ( image_3375948659692109573T_VEBT @ F @ A4 ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_VEBT_VEBT @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6427_imageE,axiom,
    ! [B: vEBT_VEBT,F: real > vEBT_VEBT,A4: set_real] :
      ( ( member_VEBT_VEBT @ B @ ( image_real_VEBT_VEBT @ F @ A4 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6428_imageE,axiom,
    ! [B: vEBT_VEBT,F: int > vEBT_VEBT,A4: set_int] :
      ( ( member_VEBT_VEBT @ B @ ( image_int_VEBT_VEBT @ F @ A4 ) )
     => ~ ! [X3: int] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_int @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6429_imageE,axiom,
    ! [B: real,F: nat > real,A4: set_nat] :
      ( ( member_real @ B @ ( image_nat_real @ F @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6430_imageE,axiom,
    ! [B: real,F: vEBT_VEBT > real,A4: set_VEBT_VEBT] :
      ( ( member_real @ B @ ( image_VEBT_VEBT_real @ F @ A4 ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_VEBT_VEBT @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_6431_image__image,axiom,
    ! [F: nat > nat,G: nat > nat,A4: set_nat] :
      ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A4 ) )
      = ( image_nat_nat
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6432_image__image,axiom,
    ! [F: nat > nat,G: int > nat,A4: set_int] :
      ( ( image_nat_nat @ F @ ( image_int_nat @ G @ A4 ) )
      = ( image_int_nat
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6433_image__image,axiom,
    ! [F: nat > int,G: nat > nat,A4: set_nat] :
      ( ( image_nat_int @ F @ ( image_nat_nat @ G @ A4 ) )
      = ( image_nat_int
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6434_image__image,axiom,
    ! [F: nat > int,G: int > nat,A4: set_int] :
      ( ( image_nat_int @ F @ ( image_int_nat @ G @ A4 ) )
      = ( image_int_int
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6435_image__image,axiom,
    ! [F: int > nat,G: nat > int,A4: set_nat] :
      ( ( image_int_nat @ F @ ( image_nat_int @ G @ A4 ) )
      = ( image_nat_nat
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6436_image__image,axiom,
    ! [F: int > nat,G: int > int,A4: set_int] :
      ( ( image_int_nat @ F @ ( image_int_int @ G @ A4 ) )
      = ( image_int_nat
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6437_image__image,axiom,
    ! [F: int > int,G: nat > int,A4: set_nat] :
      ( ( image_int_int @ F @ ( image_nat_int @ G @ A4 ) )
      = ( image_nat_int
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6438_image__image,axiom,
    ! [F: int > int,G: int > int,A4: set_int] :
      ( ( image_int_int @ F @ ( image_int_int @ G @ A4 ) )
      = ( image_int_int
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6439_image__image,axiom,
    ! [F: set_nat > nat,G: nat > set_nat,A4: set_nat] :
      ( ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G @ A4 ) )
      = ( image_nat_nat
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6440_image__image,axiom,
    ! [F: set_nat > int,G: nat > set_nat,A4: set_nat] :
      ( ( image_set_nat_int @ F @ ( image_nat_set_nat @ G @ A4 ) )
      = ( image_nat_int
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_6441_Compr__image__eq,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,A4: set_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( image_3375948659692109573T_VEBT @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_3375948659692109573T_VEBT @ F
        @ ( collect_VEBT_VEBT
          @ ^ [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6442_Compr__image__eq,axiom,
    ! [F: real > vEBT_VEBT,A4: set_real,P2: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( image_real_VEBT_VEBT @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_real_VEBT_VEBT @ F
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6443_Compr__image__eq,axiom,
    ! [F: vEBT_VEBT > real,A4: set_VEBT_VEBT,P2: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_VEBT_VEBT_real @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_VEBT_VEBT_real @ F
        @ ( collect_VEBT_VEBT
          @ ^ [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6444_Compr__image__eq,axiom,
    ! [F: real > real,A4: set_real,P2: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_real_real @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_real_real @ F
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6445_Compr__image__eq,axiom,
    ! [F: complex > vEBT_VEBT,A4: set_complex,P2: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( image_932796090930683071T_VEBT @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_932796090930683071T_VEBT @ F
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6446_Compr__image__eq,axiom,
    ! [F: complex > real,A4: set_complex,P2: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_complex_real @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_complex_real @ F
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6447_Compr__image__eq,axiom,
    ! [F: nat > vEBT_VEBT,A4: set_nat,P2: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( image_nat_VEBT_VEBT @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_nat_VEBT_VEBT @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6448_Compr__image__eq,axiom,
    ! [F: nat > real,A4: set_nat,P2: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_nat_real @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_nat_real @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6449_Compr__image__eq,axiom,
    ! [F: int > vEBT_VEBT,A4: set_int,P2: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( image_int_VEBT_VEBT @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_int_VEBT_VEBT @ F
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6450_Compr__image__eq,axiom,
    ! [F: int > real,A4: set_int,P2: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_int_real @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_int_real @ F
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6451_lambda__zero,axiom,
    ( ( ^ [H4: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_6452_lambda__zero,axiom,
    ( ( ^ [H4: uint32] : zero_zero_uint32 )
    = ( times_times_uint32 @ zero_zero_uint32 ) ) ).

% lambda_zero
thf(fact_6453_lambda__zero,axiom,
    ( ( ^ [H4: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_6454_lambda__zero,axiom,
    ( ( ^ [H4: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_6455_lambda__zero,axiom,
    ( ( ^ [H4: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_6456_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6457_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6458_sum__mono,axiom,
    ! [K4: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6459_sum__mono,axiom,
    ! [K4: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ K4 ) @ ( groups136491112297645522BT_rat @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6460_sum__mono,axiom,
    ! [K4: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K4 ) @ ( groups1300246762558778688al_rat @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6461_sum__mono,axiom,
    ! [K4: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6462_sum__mono,axiom,
    ! [K4: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ K4 ) @ ( groups771621172384141258BT_nat @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6463_sum__mono,axiom,
    ! [K4: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6464_sum__mono,axiom,
    ! [K4: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6465_sum__mono,axiom,
    ! [K4: set_nat,F: nat > int,G: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6466_sum__mono,axiom,
    ! [K4: set_VEBT_VEBT,F: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups769130701875090982BT_int @ F @ K4 ) @ ( groups769130701875090982BT_int @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6467_sum__mono,axiom,
    ! [K4: set_real,F: real > int,G: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G @ K4 ) ) ) ).

% sum_mono
thf(fact_6468_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_nat @ ( image_VEBT_VEBT_nat @ F @ A4 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A7: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6469_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite_finite_nat @ ( image_real_nat @ F @ A4 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A7: real] :
                      ( ( member_real @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6470_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > int] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_int @ ( image_VEBT_VEBT_int @ F @ A4 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A7: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6471_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: real > int] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite_finite_int @ ( image_real_int @ F @ A4 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A7: real] :
                      ( ( member_real @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6472_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > complex] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite3207457112153483333omplex @ ( image_3793382806556112285omplex @ F @ A4 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A7: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6473_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: real > complex] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite3207457112153483333omplex @ ( image_real_complex @ F @ A4 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A7: real] :
                      ( ( member_real @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6474_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite6017078050557962740nteger @ ( image_2092689629700589388nteger @ F @ A4 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A7: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6475_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: real > code_integer] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite6017078050557962740nteger @ ( image_4958697645175560720nteger @ F @ A4 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A7: real] :
                      ( ( member_real @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6476_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A7: nat] :
                      ( ( member_nat @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6477_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_int @ ( image_nat_int @ F @ A4 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A7: nat] :
                      ( ( member_nat @ A7 @ A4 )
                      & ( ( F @ A7 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6478_image__Collect__subsetI,axiom,
    ! [P2: complex > $o,F: complex > nat,B4: set_nat] :
      ( ! [X3: complex] :
          ( ( P2 @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ ( collect_complex @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6479_image__Collect__subsetI,axiom,
    ! [P2: complex > $o,F: complex > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: complex] :
          ( ( P2 @ X3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_932796090930683071T_VEBT @ F @ ( collect_complex @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6480_image__Collect__subsetI,axiom,
    ! [P2: complex > $o,F: complex > real,B4: set_real] :
      ( ! [X3: complex] :
          ( ( P2 @ X3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_complex_real @ F @ ( collect_complex @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6481_image__Collect__subsetI,axiom,
    ! [P2: nat > $o,F: nat > nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6482_image__Collect__subsetI,axiom,
    ! [P2: nat > $o,F: nat > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_nat_VEBT_VEBT @ F @ ( collect_nat @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6483_image__Collect__subsetI,axiom,
    ! [P2: nat > $o,F: nat > real,B4: set_real] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_nat_real @ F @ ( collect_nat @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6484_image__Collect__subsetI,axiom,
    ! [P2: int > $o,F: int > nat,B4: set_nat] :
      ( ! [X3: int] :
          ( ( P2 @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( collect_int @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6485_image__Collect__subsetI,axiom,
    ! [P2: int > $o,F: int > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: int] :
          ( ( P2 @ X3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_int_VEBT_VEBT @ F @ ( collect_int @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6486_image__Collect__subsetI,axiom,
    ! [P2: int > $o,F: int > real,B4: set_real] :
      ( ! [X3: int] :
          ( ( P2 @ X3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_int_real @ F @ ( collect_int @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6487_image__Collect__subsetI,axiom,
    ! [P2: complex > $o,F: complex > int,B4: set_int] :
      ( ! [X3: complex] :
          ( ( P2 @ X3 )
         => ( member_int @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_int @ ( image_complex_int @ F @ ( collect_complex @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_6488_Collect__conv__if2,axiom,
    ! [P2: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P2 @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
thf(fact_6489_Collect__conv__if2,axiom,
    ! [P2: complex > $o,A: complex] :
      ( ( ( P2 @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_6490_Collect__conv__if2,axiom,
    ! [P2: list_nat > $o,A: list_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6491_Collect__conv__if2,axiom,
    ! [P2: set_nat > $o,A: set_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6492_Collect__conv__if2,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6493_Collect__conv__if2,axiom,
    ! [P2: int > $o,A: int] :
      ( ( ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_6494_Collect__conv__if2,axiom,
    ! [P2: $o > $o,A: $o] :
      ( ( ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_6495_Collect__conv__if,axiom,
    ! [P2: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P2 @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
thf(fact_6496_Collect__conv__if,axiom,
    ! [P2: complex > $o,A: complex] :
      ( ( ( P2 @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_6497_Collect__conv__if,axiom,
    ! [P2: list_nat > $o,A: list_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_6498_Collect__conv__if,axiom,
    ! [P2: set_nat > $o,A: set_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_6499_Collect__conv__if,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_6500_Collect__conv__if,axiom,
    ! [P2: int > $o,A: int] :
      ( ( ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_6501_Collect__conv__if,axiom,
    ! [P2: $o > $o,A: $o] :
      ( ( ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_6502_sum_Oswap__restrict,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_nat,G: vEBT_VEBT > nat > nat,R2: vEBT_VEBT > nat > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups771621172384141258BT_nat
            @ ^ [X4: vEBT_VEBT] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups771621172384141258BT_nat
                @ ^ [X4: vEBT_VEBT] : ( G @ X4 @ Y4 )
                @ ( collect_VEBT_VEBT
                  @ ^ [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6503_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: real > nat > nat,R2: real > nat > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X4: real] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X4: real] : ( G @ X4 @ Y4 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6504_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,G: int > nat > nat,R2: int > nat > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X4: int] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X4: int] : ( G @ X4 @ Y4 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6505_sum_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: complex > nat > nat,R2: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X4: complex] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X4: complex] : ( G @ X4 @ Y4 )
                @ ( collect_complex
                  @ ^ [X4: complex] :
                      ( ( member_complex @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6506_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_nat,G: code_integer > nat > nat,R2: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X4: code_integer] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups7237345082560585321er_nat
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6507_sum_Oswap__restrict,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_complex,G: vEBT_VEBT > complex > complex,R2: vEBT_VEBT > complex > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups1794756597179926696omplex
            @ ^ [X4: vEBT_VEBT] :
                ( groups7754918857620584856omplex @ ( G @ X4 )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups1794756597179926696omplex
                @ ^ [X4: vEBT_VEBT] : ( G @ X4 @ Y4 )
                @ ( collect_VEBT_VEBT
                  @ ^ [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6508_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_complex,G: real > complex > complex,R2: real > complex > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5754745047067104278omplex
            @ ^ [X4: real] :
                ( groups7754918857620584856omplex @ ( G @ X4 )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X4: real] : ( G @ X4 @ Y4 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6509_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_complex,G: nat > complex > complex,R2: nat > complex > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X4: nat] :
                ( groups7754918857620584856omplex @ ( G @ X4 )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6510_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_complex,G: int > complex > complex,R2: int > complex > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X4: int] :
                ( groups7754918857620584856omplex @ ( G @ X4 )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X4: int] : ( G @ X4 @ Y4 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6511_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_complex,G: code_integer > complex > complex,R2: code_integer > complex > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups8024822376189712711omplex
            @ ^ [X4: code_integer] :
                ( groups7754918857620584856omplex @ ( G @ X4 )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups8024822376189712711omplex
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A4 )
                      & ( R2 @ X4 @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6512_sum__diff__distrib,axiom,
    ! [Q2: int > nat,P2: int > nat,N: int] :
      ( ! [X3: int] : ( ord_less_eq_nat @ ( Q2 @ X3 ) @ ( P2 @ X3 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P2 @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q2 @ ( set_ord_lessThan_int @ N ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( P2 @ X4 ) @ ( Q2 @ X4 ) )
          @ ( set_ord_lessThan_int @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_6513_sum__diff__distrib,axiom,
    ! [Q2: real > nat,P2: real > nat,N: real] :
      ( ! [X3: real] : ( ord_less_eq_nat @ ( Q2 @ X3 ) @ ( P2 @ X3 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P2 @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q2 @ ( set_or5984915006950818249n_real @ N ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( minus_minus_nat @ ( P2 @ X4 ) @ ( Q2 @ X4 ) )
          @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_6514_sum__diff__distrib,axiom,
    ! [Q2: nat > nat,P2: nat > nat,N: nat] :
      ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q2 @ X3 ) @ ( P2 @ X3 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P2 @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q2 @ ( set_ord_lessThan_nat @ N ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( P2 @ X4 ) @ ( Q2 @ X4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_6515_max__def__raw,axiom,
    ( ord_max_Code_integer
    = ( ^ [A7: code_integer,B7: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_6516_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A7: set_int,B7: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_6517_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A7: rat,B7: rat] : ( if_rat @ ( ord_less_eq_rat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_6518_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A7: num,B7: num] : ( if_num @ ( ord_less_eq_num @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_6519_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A7: nat,B7: nat] : ( if_nat @ ( ord_less_eq_nat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_6520_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A7: int,B7: int] : ( if_int @ ( ord_less_eq_int @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_6521_pred__subset__eq2,axiom,
    ! [R2: set_Pr1773385645901665561uint32,S3: set_Pr1773385645901665561uint32] :
      ( ( ord_le5999336846926846542nt32_o
        @ ^ [X4: uint32,Y4: uint32] : ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) @ R2 )
        @ ^ [X4: uint32,Y4: uint32] : ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) @ S3 ) )
      = ( ord_le6429528607962791097uint32 @ R2 @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6522_pred__subset__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R2 )
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S3 ) )
      = ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6523_pred__subset__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R2 )
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ S3 ) )
      = ( ord_le2843351958646193337nt_int @ R2 @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6524_pred__subset__eq2,axiom,
    ! [R2: set_Pr3286484037609594932et_nat,S3: set_Pr3286484037609594932et_nat] :
      ( ( ord_le8000401564054156549_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R2 )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S3 ) )
      = ( ord_le5966269811547037844et_nat @ R2 @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6525_pred__subset__eq2,axiom,
    ! [R2: set_Pr8536935166611901872et_nat,S3: set_Pr8536935166611901872et_nat] :
      ( ( ord_le6753239538765779593_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R2 )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S3 ) )
      = ( ord_le4763372923235995152et_nat @ R2 @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6526_bot__empty__eq2,axiom,
    ( bot_bo8112279147790955290nt32_o
    = ( ^ [X4: uint32,Y4: uint32] : ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) @ bot_bo8438649754162204037uint32 ) ) ) ).

% bot_empty_eq2
thf(fact_6527_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6528_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6529_bot__empty__eq2,axiom,
    ( bot_bo5580076615179976505_nat_o
    = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ bot_bo1481135142794719944et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6530_bot__empty__eq2,axiom,
    ( bot_bo3790638025767943357_nat_o
    = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ bot_bo5635537948650799172et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6531_finite__M__bounded__by__nat,axiom,
    ! [P2: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P2 @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_6532_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N6: nat] : ( ord_less_eq_nat @ ( F @ N6 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_6533_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_6534_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_VEBT_VEBT,A2: list_VEBT_VEBT,B2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_s6755466524823107622T_VEBT @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_s6755466524823107622T_VEBT @ B2 ) )
       => ( ( ( zip_VE537291747668921783T_VEBT @ A @ B )
            = ( zip_VE537291747668921783T_VEBT @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6535_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_real,A2: list_VEBT_VEBT,B2: list_real] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_size_list_real @ B2 ) )
       => ( ( ( zip_VEBT_VEBT_real @ A @ B )
            = ( zip_VEBT_VEBT_real @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6536_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_o,A2: list_VEBT_VEBT,B2: list_o] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_VEBT_VEBT_o @ A @ B )
            = ( zip_VEBT_VEBT_o @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6537_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_nat,A2: list_VEBT_VEBT,B2: list_nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_VEBT_VEBT_nat @ A @ B )
            = ( zip_VEBT_VEBT_nat @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6538_zip__inj,axiom,
    ! [A: list_real,B: list_VEBT_VEBT,A2: list_real,B2: list_VEBT_VEBT] :
      ( ( ( size_size_list_real @ A )
        = ( size_s6755466524823107622T_VEBT @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_s6755466524823107622T_VEBT @ B2 ) )
       => ( ( ( zip_real_VEBT_VEBT @ A @ B )
            = ( zip_real_VEBT_VEBT @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6539_zip__inj,axiom,
    ! [A: list_real,B: list_real,A2: list_real,B2: list_real] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_size_list_real @ B2 ) )
       => ( ( ( zip_real_real @ A @ B )
            = ( zip_real_real @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6540_zip__inj,axiom,
    ! [A: list_real,B: list_o,A2: list_real,B2: list_o] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_real_o @ A @ B )
            = ( zip_real_o @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6541_zip__inj,axiom,
    ! [A: list_real,B: list_nat,A2: list_real,B2: list_nat] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_real @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_real_nat @ A @ B )
            = ( zip_real_nat @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6542_zip__inj,axiom,
    ! [A: list_o,B: list_VEBT_VEBT,A2: list_o,B2: list_VEBT_VEBT] :
      ( ( ( size_size_list_o @ A )
        = ( size_s6755466524823107622T_VEBT @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_s6755466524823107622T_VEBT @ B2 ) )
       => ( ( ( zip_o_VEBT_VEBT @ A @ B )
            = ( zip_o_VEBT_VEBT @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6543_zip__inj,axiom,
    ! [A: list_o,B: list_real,A2: list_o,B2: list_real] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_real @ B2 ) )
       => ( ( ( zip_o_real @ A @ B )
            = ( zip_o_real @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_6544_pair__list__split,axiom,
    ! [L: list_P7413028617227757229T_VEBT] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_VEBT_VEBT] :
          ( ( L
            = ( zip_VE537291747668921783T_VEBT @ L1 @ L22 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ L1 )
              = ( size_s6755466524823107622T_VEBT @ L22 ) )
           => ( ( size_s7466405169056248089T_VEBT @ L )
             != ( size_s6755466524823107622T_VEBT @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6545_pair__list__split,axiom,
    ! [L: list_P2623026923184700063T_real] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_real] :
          ( ( L
            = ( zip_VEBT_VEBT_real @ L1 @ L22 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ L1 )
              = ( size_size_list_real @ L22 ) )
           => ( ( size_s5035110155006384947T_real @ L )
             != ( size_size_list_real @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6546_pair__list__split,axiom,
    ! [L: list_P3126845725202233233VEBT_o] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_o] :
          ( ( L
            = ( zip_VEBT_VEBT_o @ L1 @ L22 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ L1 )
              = ( size_size_list_o @ L22 ) )
           => ( ( size_s9168528473962070013VEBT_o @ L )
             != ( size_size_list_o @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6547_pair__list__split,axiom,
    ! [L: list_P7037539587688870467BT_nat] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_nat] :
          ( ( L
            = ( zip_VEBT_VEBT_nat @ L1 @ L22 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ L1 )
              = ( size_size_list_nat @ L22 ) )
           => ( ( size_s6152045936467909847BT_nat @ L )
             != ( size_size_list_nat @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6548_pair__list__split,axiom,
    ! [L: list_P877281246627933069T_VEBT] :
      ~ ! [L1: list_real,L22: list_VEBT_VEBT] :
          ( ( L
            = ( zip_real_VEBT_VEBT @ L1 @ L22 ) )
         => ( ( ( size_size_list_real @ L1 )
              = ( size_s6755466524823107622T_VEBT @ L22 ) )
           => ( ( size_s3289364478449617953T_VEBT @ L )
             != ( size_s6755466524823107622T_VEBT @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6549_pair__list__split,axiom,
    ! [L: list_P8689742595348180415l_real] :
      ~ ! [L1: list_real,L22: list_real] :
          ( ( L
            = ( zip_real_real @ L1 @ L22 ) )
         => ( ( ( size_size_list_real @ L1 )
              = ( size_size_list_real @ L22 ) )
           => ( ( size_s3932428310213730859l_real @ L )
             != ( size_size_list_real @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6550_pair__list__split,axiom,
    ! [L: list_P3595434254542482545real_o] :
      ~ ! [L1: list_real,L22: list_o] :
          ( ( L
            = ( zip_real_o @ L1 @ L22 ) )
         => ( ( ( size_size_list_real @ L1 )
              = ( size_size_list_o @ L22 ) )
           => ( ( size_s987546567493390085real_o @ L )
             != ( size_size_list_o @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6551_pair__list__split,axiom,
    ! [L: list_P6834414599653733731al_nat] :
      ~ ! [L1: list_real,L22: list_nat] :
          ( ( L
            = ( zip_real_nat @ L1 @ L22 ) )
         => ( ( ( size_size_list_real @ L1 )
              = ( size_size_list_nat @ L22 ) )
           => ( ( size_s1877336372972134351al_nat @ L )
             != ( size_size_list_nat @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6552_pair__list__split,axiom,
    ! [L: list_P7495141550334521929T_VEBT] :
      ~ ! [L1: list_o,L22: list_VEBT_VEBT] :
          ( ( L
            = ( zip_o_VEBT_VEBT @ L1 @ L22 ) )
         => ( ( ( size_size_list_o @ L1 )
              = ( size_s6755466524823107622T_VEBT @ L22 ) )
           => ( ( size_s4313452262239582901T_VEBT @ L )
             != ( size_s6755466524823107622T_VEBT @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6553_pair__list__split,axiom,
    ! [L: list_P5232166724548748803o_real] :
      ~ ! [L1: list_o,L22: list_real] :
          ( ( L
            = ( zip_o_real @ L1 @ L22 ) )
         => ( ( ( size_size_list_o @ L1 )
              = ( size_size_list_real @ L22 ) )
           => ( ( size_s2624279037499656343o_real @ L )
             != ( size_size_list_real @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_6554_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% real_arch_simple
thf(fact_6555_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% real_arch_simple
thf(fact_6556_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% reals_Archimedean2
thf(fact_6557_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% reals_Archimedean2
thf(fact_6558_mult__of__nat__commute,axiom,
    ! [X: nat,Y: uint32] :
      ( ( times_times_uint32 @ ( semiri2565882477558803405uint32 @ X ) @ Y )
      = ( times_times_uint32 @ Y @ ( semiri2565882477558803405uint32 @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_6559_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_6560_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_6561_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_6562_mult__of__nat__commute,axiom,
    ! [X: nat,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
      = ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_6563_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_6564_lessThan__non__empty,axiom,
    ! [X: real] :
      ( ( set_or5984915006950818249n_real @ X )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_6565_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_6566_infinite__Iio,axiom,
    ! [A: real] :
      ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).

% infinite_Iio
thf(fact_6567_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6568_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6569_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6570_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6571_sum__lessThan__telescope,axiom,
    ! [F: nat > uint32,M: nat] :
      ( ( groups833757482993574392uint32
        @ ^ [N6: nat] : ( minus_minus_uint32 @ ( F @ ( suc @ N6 ) ) @ ( F @ N6 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_uint32 @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6572_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N6: nat] : ( minus_minus_int @ ( F @ ( suc @ N6 ) ) @ ( F @ N6 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6573_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N6: nat] : ( minus_minus_real @ ( F @ ( suc @ N6 ) ) @ ( F @ N6 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6574_sum__lessThan__telescope_H,axiom,
    ! [F: nat > uint32,M: nat] :
      ( ( groups833757482993574392uint32
        @ ^ [N6: nat] : ( minus_minus_uint32 @ ( F @ N6 ) @ ( F @ ( suc @ N6 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_uint32 @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6575_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N6: nat] : ( minus_minus_int @ ( F @ N6 ) @ ( F @ ( suc @ N6 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6576_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N6: nat] : ( minus_minus_real @ ( F @ N6 ) @ ( F @ ( suc @ N6 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6577_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > uint32,Y: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6578_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > uint32,Y: real > uint32] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6579_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > uint32,Y: nat > uint32] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6580_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > uint32,Y: int > uint32] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6581_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > uint32,Y: complex > uint32] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6582_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > uint32,Y: code_integer > uint32] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6583_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > real,Y: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6584_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6585_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6586_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6587_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > rat,Y: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6588_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6589_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6590_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6591_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > rat,Y: complex > rat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6592_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > rat,Y: code_integer > rat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6593_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > assn,Y: vEBT_VEBT > assn] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6594_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > assn,Y: real > assn] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6595_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > assn,Y: nat > assn] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6596_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > assn,Y: int > assn] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6597_filter__preserves__multiset,axiom,
    ! [M7: list_nat > nat,P2: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X4 ) @ ( M7 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6598_filter__preserves__multiset,axiom,
    ! [M7: set_nat > nat,P2: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X4 ) @ ( M7 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6599_filter__preserves__multiset,axiom,
    ! [M7: nat > nat,P2: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X4 ) @ ( M7 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6600_filter__preserves__multiset,axiom,
    ! [M7: int > nat,P2: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X4 ) @ ( M7 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6601_filter__preserves__multiset,axiom,
    ! [M7: complex > nat,P2: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X4 ) @ ( M7 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6602_filter__preserves__multiset,axiom,
    ! [M7: code_integer > nat,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X4 ) @ ( M7 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6603_sum_Ointer__filter,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > uint32,P2: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups8325533452322294502uint32 @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8325533452322294502uint32
          @ ^ [X4: vEBT_VEBT] : ( if_uint32 @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_uint32 )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6604_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: real > uint32,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups5944083974425963860uint32 @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups5944083974425963860uint32
          @ ^ [X4: real] : ( if_uint32 @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_uint32 )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6605_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > uint32,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups833757482993574392uint32 @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups833757482993574392uint32
          @ ^ [X4: nat] : ( if_uint32 @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_uint32 )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6606_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: int > uint32,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups5712668689793887828uint32 @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups5712668689793887828uint32
          @ ^ [X4: int] : ( if_uint32 @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_uint32 )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6607_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: complex > uint32,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups8736914816313324502uint32 @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8736914816313324502uint32
          @ ^ [X4: complex] : ( if_uint32 @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_uint32 )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6608_sum_Ointer__filter,axiom,
    ! [A4: set_Code_integer,G: code_integer > uint32,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups8847630953604152069uint32 @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8847630953604152069uint32
          @ ^ [X4: code_integer] : ( if_uint32 @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_uint32 )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6609_sum_Ointer__filter,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > real,P2: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups2240296850493347238T_real @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups2240296850493347238T_real
          @ ^ [X4: vEBT_VEBT] : ( if_real @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6610_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: real > real,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X4: real] : ( if_real @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6611_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: int > real,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X4: int] : ( if_real @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6612_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: complex > real,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X4: complex] : ( if_real @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_6613_image__constant,axiom,
    ! [X: nat,A4: set_nat,C2: vEBT_VEBT] :
      ( ( member_nat @ X @ A4 )
     => ( ( image_nat_VEBT_VEBT
          @ ^ [X4: nat] : C2
          @ A4 )
        = ( insert_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_6614_image__constant,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( image_3375948659692109573T_VEBT
          @ ^ [X4: vEBT_VEBT] : C2
          @ A4 )
        = ( insert_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_6615_image__constant,axiom,
    ! [X: real,A4: set_real,C2: vEBT_VEBT] :
      ( ( member_real @ X @ A4 )
     => ( ( image_real_VEBT_VEBT
          @ ^ [X4: real] : C2
          @ A4 )
        = ( insert_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_6616_image__constant,axiom,
    ! [X: int,A4: set_int,C2: vEBT_VEBT] :
      ( ( member_int @ X @ A4 )
     => ( ( image_int_VEBT_VEBT
          @ ^ [X4: int] : C2
          @ A4 )
        = ( insert_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_6617_image__constant,axiom,
    ! [X: nat,A4: set_nat,C2: nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( image_nat_nat
          @ ^ [X4: nat] : C2
          @ A4 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_6618_image__constant,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,C2: nat] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( image_VEBT_VEBT_nat
          @ ^ [X4: vEBT_VEBT] : C2
          @ A4 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_6619_image__constant,axiom,
    ! [X: real,A4: set_real,C2: nat] :
      ( ( member_real @ X @ A4 )
     => ( ( image_real_nat
          @ ^ [X4: real] : C2
          @ A4 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_6620_image__constant,axiom,
    ! [X: int,A4: set_int,C2: nat] :
      ( ( member_int @ X @ A4 )
     => ( ( image_int_nat
          @ ^ [X4: int] : C2
          @ A4 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_6621_image__constant,axiom,
    ! [X: nat,A4: set_nat,C2: int] :
      ( ( member_nat @ X @ A4 )
     => ( ( image_nat_int
          @ ^ [X4: nat] : C2
          @ A4 )
        = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ).

% image_constant
thf(fact_6622_image__constant,axiom,
    ! [X: vEBT_VEBT,A4: set_VEBT_VEBT,C2: int] :
      ( ( member_VEBT_VEBT @ X @ A4 )
     => ( ( image_VEBT_VEBT_int
          @ ^ [X4: vEBT_VEBT] : C2
          @ A4 )
        = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ).

% image_constant
thf(fact_6623_image__constant__conv,axiom,
    ! [A4: set_nat,C2: vEBT_VEBT] :
      ( ( ( A4 = bot_bot_set_nat )
       => ( ( image_nat_VEBT_VEBT
            @ ^ [X4: nat] : C2
            @ A4 )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( A4 != bot_bot_set_nat )
       => ( ( image_nat_VEBT_VEBT
            @ ^ [X4: nat] : C2
            @ A4 )
          = ( insert_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% image_constant_conv
thf(fact_6624_image__constant__conv,axiom,
    ! [A4: set_nat,C2: nat] :
      ( ( ( A4 = bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X4: nat] : C2
            @ A4 )
          = bot_bot_set_nat ) )
      & ( ( A4 != bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X4: nat] : C2
            @ A4 )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_6625_image__constant__conv,axiom,
    ! [A4: set_nat,C2: int] :
      ( ( ( A4 = bot_bot_set_nat )
       => ( ( image_nat_int
            @ ^ [X4: nat] : C2
            @ A4 )
          = bot_bot_set_int ) )
      & ( ( A4 != bot_bot_set_nat )
       => ( ( image_nat_int
            @ ^ [X4: nat] : C2
            @ A4 )
          = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ) ).

% image_constant_conv
thf(fact_6626_image__constant__conv,axiom,
    ! [A4: set_nat,C2: $o] :
      ( ( ( A4 = bot_bot_set_nat )
       => ( ( image_nat_o
            @ ^ [X4: nat] : C2
            @ A4 )
          = bot_bot_set_o ) )
      & ( ( A4 != bot_bot_set_nat )
       => ( ( image_nat_o
            @ ^ [X4: nat] : C2
            @ A4 )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_6627_image__constant__conv,axiom,
    ! [A4: set_int,C2: vEBT_VEBT] :
      ( ( ( A4 = bot_bot_set_int )
       => ( ( image_int_VEBT_VEBT
            @ ^ [X4: int] : C2
            @ A4 )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( A4 != bot_bot_set_int )
       => ( ( image_int_VEBT_VEBT
            @ ^ [X4: int] : C2
            @ A4 )
          = ( insert_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% image_constant_conv
thf(fact_6628_image__constant__conv,axiom,
    ! [A4: set_int,C2: nat] :
      ( ( ( A4 = bot_bot_set_int )
       => ( ( image_int_nat
            @ ^ [X4: int] : C2
            @ A4 )
          = bot_bot_set_nat ) )
      & ( ( A4 != bot_bot_set_int )
       => ( ( image_int_nat
            @ ^ [X4: int] : C2
            @ A4 )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_6629_image__constant__conv,axiom,
    ! [A4: set_int,C2: int] :
      ( ( ( A4 = bot_bot_set_int )
       => ( ( image_int_int
            @ ^ [X4: int] : C2
            @ A4 )
          = bot_bot_set_int ) )
      & ( ( A4 != bot_bot_set_int )
       => ( ( image_int_int
            @ ^ [X4: int] : C2
            @ A4 )
          = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ) ).

% image_constant_conv
thf(fact_6630_image__constant__conv,axiom,
    ! [A4: set_int,C2: $o] :
      ( ( ( A4 = bot_bot_set_int )
       => ( ( image_int_o
            @ ^ [X4: int] : C2
            @ A4 )
          = bot_bot_set_o ) )
      & ( ( A4 != bot_bot_set_int )
       => ( ( image_int_o
            @ ^ [X4: int] : C2
            @ A4 )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_6631_image__constant__conv,axiom,
    ! [A4: set_o,C2: vEBT_VEBT] :
      ( ( ( A4 = bot_bot_set_o )
       => ( ( image_o_VEBT_VEBT
            @ ^ [X4: $o] : C2
            @ A4 )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( A4 != bot_bot_set_o )
       => ( ( image_o_VEBT_VEBT
            @ ^ [X4: $o] : C2
            @ A4 )
          = ( insert_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% image_constant_conv
thf(fact_6632_image__constant__conv,axiom,
    ! [A4: set_o,C2: nat] :
      ( ( ( A4 = bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X4: $o] : C2
            @ A4 )
          = bot_bot_set_nat ) )
      & ( ( A4 != bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X4: $o] : C2
            @ A4 )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_6633_foldl__length__aux,axiom,
    ! [A: nat,L: list_VEBT_VEBT] :
      ( ( foldl_nat_VEBT_VEBT
        @ ^ [I3: nat,X4: vEBT_VEBT] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_s6755466524823107622T_VEBT @ L ) ) ) ).

% foldl_length_aux
thf(fact_6634_foldl__length__aux,axiom,
    ! [A: nat,L: list_real] :
      ( ( foldl_nat_real
        @ ^ [I3: nat,X4: real] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_size_list_real @ L ) ) ) ).

% foldl_length_aux
thf(fact_6635_foldl__length__aux,axiom,
    ! [A: nat,L: list_o] :
      ( ( foldl_nat_o
        @ ^ [I3: nat,X4: $o] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_size_list_o @ L ) ) ) ).

% foldl_length_aux
thf(fact_6636_foldl__length__aux,axiom,
    ! [A: nat,L: list_nat] :
      ( ( foldl_nat_nat
        @ ^ [I3: nat,X4: nat] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_size_list_nat @ L ) ) ) ).

% foldl_length_aux
thf(fact_6637_sum_Oimage__gen,axiom,
    ! [S3: set_VEBT_VEBT,H2: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( groups771621172384141258BT_nat @ H2 @ S3 )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups771621172384141258BT_nat @ H2
              @ ( collect_VEBT_VEBT
                @ ^ [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_VEBT_VEBT_nat @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6638_sum_Oimage__gen,axiom,
    ! [S3: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( groups1935376822645274424al_nat @ H2 @ S3 )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups1935376822645274424al_nat @ H2
              @ ( collect_real
                @ ^ [X4: real] :
                    ( ( member_real @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_real_nat @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6639_sum_Oimage__gen,axiom,
    ! [S3: set_int,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( groups4541462559716669496nt_nat @ H2 @ S3 )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups4541462559716669496nt_nat @ H2
              @ ( collect_int
                @ ^ [X4: int] :
                    ( ( member_int @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_int_nat @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6640_sum_Oimage__gen,axiom,
    ! [S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( groups5693394587270226106ex_nat @ H2 @ S3 )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups5693394587270226106ex_nat @ H2
              @ ( collect_complex
                @ ^ [X4: complex] :
                    ( ( member_complex @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_complex_nat @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6641_sum_Oimage__gen,axiom,
    ! [S3: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( groups7237345082560585321er_nat @ H2 @ S3 )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups7237345082560585321er_nat @ H2
              @ ( collect_Code_integer
                @ ^ [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_951025933927791156er_nat @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6642_sum_Oimage__gen,axiom,
    ! [S3: set_VEBT_VEBT,H2: vEBT_VEBT > complex,G: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( groups1794756597179926696omplex @ H2 @ S3 )
        = ( groups7754918857620584856omplex
          @ ^ [Y4: complex] :
              ( groups1794756597179926696omplex @ H2
              @ ( collect_VEBT_VEBT
                @ ^ [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_3793382806556112285omplex @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6643_sum_Oimage__gen,axiom,
    ! [S3: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( groups5754745047067104278omplex @ H2 @ S3 )
        = ( groups7754918857620584856omplex
          @ ^ [Y4: complex] :
              ( groups5754745047067104278omplex @ H2
              @ ( collect_real
                @ ^ [X4: real] :
                    ( ( member_real @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_real_complex @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6644_sum_Oimage__gen,axiom,
    ! [S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( groups2073611262835488442omplex @ H2 @ S3 )
        = ( groups7754918857620584856omplex
          @ ^ [Y4: complex] :
              ( groups2073611262835488442omplex @ H2
              @ ( collect_nat
                @ ^ [X4: nat] :
                    ( ( member_nat @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_nat_complex @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6645_sum_Oimage__gen,axiom,
    ! [S3: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( groups3049146728041665814omplex @ H2 @ S3 )
        = ( groups7754918857620584856omplex
          @ ^ [Y4: complex] :
              ( groups3049146728041665814omplex @ H2
              @ ( collect_int
                @ ^ [X4: int] :
                    ( ( member_int @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_int_complex @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6646_sum_Oimage__gen,axiom,
    ! [S3: set_Code_integer,H2: code_integer > complex,G: code_integer > complex] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( groups8024822376189712711omplex @ H2 @ S3 )
        = ( groups7754918857620584856omplex
          @ ^ [Y4: complex] :
              ( groups8024822376189712711omplex @ H2
              @ ( collect_Code_integer
                @ ^ [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ S3 )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_3397630267976458002omplex @ G @ S3 ) ) ) ) ).

% sum.image_gen
thf(fact_6647_nat__seg__image__imp__finite,axiom,
    ! [A4: set_set_nat,F: nat > set_nat,N: nat] :
      ( ( A4
        = ( image_nat_set_nat @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite1152437895449049373et_nat @ A4 ) ) ).

% nat_seg_image_imp_finite
thf(fact_6648_nat__seg__image__imp__finite,axiom,
    ! [A4: set_nat,F: nat > nat,N: nat] :
      ( ( A4
        = ( image_nat_nat @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite_finite_nat @ A4 ) ) ).

% nat_seg_image_imp_finite
thf(fact_6649_nat__seg__image__imp__finite,axiom,
    ! [A4: set_int,F: nat > int,N: nat] :
      ( ( A4
        = ( image_nat_int @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite_finite_int @ A4 ) ) ).

% nat_seg_image_imp_finite
thf(fact_6650_nat__seg__image__imp__finite,axiom,
    ! [A4: set_complex,F: nat > complex,N: nat] :
      ( ( A4
        = ( image_nat_complex @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite3207457112153483333omplex @ A4 ) ) ).

% nat_seg_image_imp_finite
thf(fact_6651_nat__seg__image__imp__finite,axiom,
    ! [A4: set_Code_integer,F: nat > code_integer,N: nat] :
      ( ( A4
        = ( image_1215581382706833972nteger @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite6017078050557962740nteger @ A4 ) ) ).

% nat_seg_image_imp_finite
thf(fact_6652_finite__conv__nat__seg__image,axiom,
    ( finite1152437895449049373et_nat
    = ( ^ [A6: set_set_nat] :
        ? [N6: nat,F5: nat > set_nat] :
          ( A6
          = ( image_nat_set_nat @ F5
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N6 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_6653_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A6: set_nat] :
        ? [N6: nat,F5: nat > nat] :
          ( A6
          = ( image_nat_nat @ F5
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N6 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_6654_finite__conv__nat__seg__image,axiom,
    ( finite_finite_int
    = ( ^ [A6: set_int] :
        ? [N6: nat,F5: nat > int] :
          ( A6
          = ( image_nat_int @ F5
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N6 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_6655_finite__conv__nat__seg__image,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A6: set_complex] :
        ? [N6: nat,F5: nat > complex] :
          ( A6
          = ( image_nat_complex @ F5
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N6 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_6656_finite__conv__nat__seg__image,axiom,
    ( finite6017078050557962740nteger
    = ( ^ [A6: set_Code_integer] :
        ? [N6: nat,F5: nat > code_integer] :
          ( A6
          = ( image_1215581382706833972nteger @ F5
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N6 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_6657_finite__int__segment,axiom,
    ! [A: real,B: real] :
      ( finite_finite_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ring_1_Ints_real )
            & ( ord_less_eq_real @ A @ X4 )
            & ( ord_less_eq_real @ X4 @ B ) ) ) ) ).

% finite_int_segment
thf(fact_6658_finite__int__segment,axiom,
    ! [A: rat,B: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [X4: rat] :
            ( ( member_rat @ X4 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ A @ X4 )
            & ( ord_less_eq_rat @ X4 @ B ) ) ) ) ).

% finite_int_segment
thf(fact_6659_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_6660_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_6661_sum__subtractf__nat,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > nat,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups771621172384141258BT_nat
          @ ^ [X4: vEBT_VEBT] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) @ ( groups771621172384141258BT_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6662_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: real > nat,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6663_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: int > nat,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6664_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X4: set_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( groups8294997508430121362at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6665_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6666_sum__multicount__gen,axiom,
    ! [S: set_VEBT_VEBT,T: set_VEBT_VEBT,R2: vEBT_VEBT > vEBT_VEBT > $o,K: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ( finite5795047828879050333T_VEBT @ T )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ T )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [J3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups771621172384141258BT_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6667_sum__multicount__gen,axiom,
    ! [S: set_VEBT_VEBT,T: set_real,R2: vEBT_VEBT > real > $o,K: real > nat] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ( finite_finite_real @ T )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6668_sum__multicount__gen,axiom,
    ! [S: set_real,T: set_VEBT_VEBT,R2: real > vEBT_VEBT > $o,K: vEBT_VEBT > nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite5795047828879050333T_VEBT @ T )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I3: real] :
                        ( ( member_real @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I3: real] :
                  ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [J3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups771621172384141258BT_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6669_sum__multicount__gen,axiom,
    ! [S: set_real,T: set_real,R2: real > real > $o,K: real > nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite_finite_real @ T )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I3: real] :
                        ( ( member_real @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I3: real] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6670_sum__multicount__gen,axiom,
    ! [S: set_VEBT_VEBT,T: set_literal,R2: vEBT_VEBT > literal > $o,K: literal > nat] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ( finite5847741373460823677iteral @ T )
       => ( ! [X3: literal] :
              ( ( member_literal @ X3 @ T )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [J3: literal] :
                        ( ( member_literal @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups8652099787943017962al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6671_sum__multicount__gen,axiom,
    ! [S: set_real,T: set_literal,R2: real > literal > $o,K: literal > nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite5847741373460823677iteral @ T )
       => ( ! [X3: literal] :
              ( ( member_literal @ X3 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I3: real] :
                        ( ( member_real @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I3: real] :
                  ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [J3: literal] :
                        ( ( member_literal @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups8652099787943017962al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6672_sum__multicount__gen,axiom,
    ! [S: set_literal,T: set_VEBT_VEBT,R2: literal > vEBT_VEBT > $o,K: vEBT_VEBT > nat] :
      ( ( finite5847741373460823677iteral @ S )
     => ( ( finite5795047828879050333T_VEBT @ T )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ T )
             => ( ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [I3: literal] :
                        ( ( member_literal @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8652099787943017962al_nat
              @ ^ [I3: literal] :
                  ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [J3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups771621172384141258BT_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6673_sum__multicount__gen,axiom,
    ! [S: set_literal,T: set_real,R2: literal > real > $o,K: real > nat] :
      ( ( finite5847741373460823677iteral @ S )
     => ( ( finite_finite_real @ T )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T )
             => ( ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [I3: literal] :
                        ( ( member_literal @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8652099787943017962al_nat
              @ ^ [I3: literal] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6674_sum__multicount__gen,axiom,
    ! [S: set_literal,T: set_literal,R2: literal > literal > $o,K: literal > nat] :
      ( ( finite5847741373460823677iteral @ S )
     => ( ( finite5847741373460823677iteral @ T )
       => ( ! [X3: literal] :
              ( ( member_literal @ X3 @ T )
             => ( ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [I3: literal] :
                        ( ( member_literal @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8652099787943017962al_nat
              @ ^ [I3: literal] :
                  ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [J3: literal] :
                        ( ( member_literal @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups8652099787943017962al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6675_sum__multicount__gen,axiom,
    ! [S: set_VEBT_VEBT,T: set_int,R2: vEBT_VEBT > int > $o,K: int > nat] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S )
            = ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_6676_total__onI,axiom,
    ! [A4: set_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ( member_VEBT_VEBT @ Y3 @ A4 )
           => ( ( X3 != Y3 )
             => ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y3 ) @ R )
                | ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_VEBT_VEBT @ A4 @ R ) ) ).

% total_onI
thf(fact_6677_total__onI,axiom,
    ! [A4: set_real,R: set_Pr6218003697084177305l_real] :
      ( ! [X3: real,Y3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ( member_real @ Y3 @ A4 )
           => ( ( X3 != Y3 )
             => ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X3 @ Y3 ) @ R )
                | ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_real @ A4 @ R ) ) ).

% total_onI
thf(fact_6678_total__onI,axiom,
    ! [A4: set_set_nat,R: set_Pr5488025237498180813et_nat] :
      ( ! [X3: set_nat,Y3: set_nat] :
          ( ( member_set_nat @ X3 @ A4 )
         => ( ( member_set_nat @ Y3 @ A4 )
           => ( ( X3 != Y3 )
             => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y3 ) @ R )
                | ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_set_nat @ A4 @ R ) ) ).

% total_onI
thf(fact_6679_total__onI,axiom,
    ! [A4: set_uint32,R: set_Pr1773385645901665561uint32] :
      ( ! [X3: uint32,Y3: uint32] :
          ( ( member_uint32 @ X3 @ A4 )
         => ( ( member_uint32 @ Y3 @ A4 )
           => ( ( X3 != Y3 )
             => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ Y3 ) @ R )
                | ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_uint32 @ A4 @ R ) ) ).

% total_onI
thf(fact_6680_total__onI,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( X3 != Y3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_nat @ A4 @ R ) ) ).

% total_onI
thf(fact_6681_total__onI,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( member_int @ Y3 @ A4 )
           => ( ( X3 != Y3 )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
                | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_int @ A4 @ R ) ) ).

% total_onI
thf(fact_6682_total__on__def,axiom,
    ( total_on_uint32
    = ( ^ [A6: set_uint32,R5: set_Pr1773385645901665561uint32] :
        ! [X4: uint32] :
          ( ( member_uint32 @ X4 @ A6 )
         => ! [Y4: uint32] :
              ( ( member_uint32 @ Y4 @ A6 )
             => ( ( X4 != Y4 )
               => ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X4 @ Y4 ) @ R5 )
                  | ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ Y4 @ X4 ) @ R5 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_6683_total__on__def,axiom,
    ( total_on_nat
    = ( ^ [A6: set_nat,R5: set_Pr1261947904930325089at_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A6 )
             => ( ( X4 != Y4 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R5 )
                  | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X4 ) @ R5 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_6684_total__on__def,axiom,
    ( total_on_int
    = ( ^ [A6: set_int,R5: set_Pr958786334691620121nt_int] :
        ! [X4: int] :
          ( ( member_int @ X4 @ A6 )
         => ! [Y4: int] :
              ( ( member_int @ Y4 @ A6 )
             => ( ( X4 != Y4 )
               => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R5 )
                  | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ X4 ) @ R5 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_6685_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_int,S2: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S8: set_int] :
              ( ( minus_minus_set_int @ S8 @ S3 )
              = ( minus_minus_set_int @ S2 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6686_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_complex,S2: set_complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [S8: set_complex] :
              ( ( minus_811609699411566653omplex @ S8 @ S3 )
              = ( minus_811609699411566653omplex @ S2 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6687_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_Code_integer,S2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S8: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S8 @ S3 )
              = ( minus_2355218937544613996nteger @ S2 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6688_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_nat,S2: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S8: set_nat] :
              ( ( minus_minus_set_nat @ S8 @ S3 )
              = ( minus_minus_set_nat @ S2 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6689_total__on__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( total_on_nat @ bot_bot_set_nat @ R ) ).

% total_on_empty
thf(fact_6690_total__on__empty,axiom,
    ! [R: set_Pr958786334691620121nt_int] : ( total_on_int @ bot_bot_set_int @ R ) ).

% total_on_empty
thf(fact_6691_total__on__empty,axiom,
    ! [R: set_Product_prod_o_o] : ( total_on_o @ bot_bot_set_o @ R ) ).

% total_on_empty
thf(fact_6692_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A6: set_nat] :
          ( collect_set_nat
          @ ^ [X8: set_nat] :
              ( ( ord_less_eq_set_nat @ X8 @ A6 )
              & ( finite_finite_nat @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_6693_Fpow__def,axiom,
    ( finite_Fpow_complex
    = ( ^ [A6: set_complex] :
          ( collect_set_complex
          @ ^ [X8: set_complex] :
              ( ( ord_le211207098394363844omplex @ X8 @ A6 )
              & ( finite3207457112153483333omplex @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_6694_Fpow__def,axiom,
    ( finite1532502677820914807nteger
    = ( ^ [A6: set_Code_integer] :
          ( collec574505750873337192nteger
          @ ^ [X8: set_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ X8 @ A6 )
              & ( finite6017078050557962740nteger @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_6695_Fpow__def,axiom,
    ( finite_Fpow_int
    = ( ^ [A6: set_int] :
          ( collect_set_int
          @ ^ [X8: set_int] :
              ( ( ord_less_eq_set_int @ X8 @ A6 )
              & ( finite_finite_int @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_6696_distinct__finite__set,axiom,
    ! [X: set_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT
      @ ( collec5608196760682091941T_VEBT
        @ ^ [Ys3: list_VEBT_VEBT] :
            ( ( ( set_VEBT_VEBT2 @ Ys3 )
              = X )
            & ( distinct_VEBT_VEBT @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_6697_distinct__finite__set,axiom,
    ! [X: set_nat] :
      ( finite8100373058378681591st_nat
      @ ( collect_list_nat
        @ ^ [Ys3: list_nat] :
            ( ( ( set_nat2 @ Ys3 )
              = X )
            & ( distinct_nat @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_6698_distinct__finite__set,axiom,
    ! [X: set_real] :
      ( finite306553202115118035t_real
      @ ( collect_list_real
        @ ^ [Ys3: list_real] :
            ( ( ( set_real2 @ Ys3 )
              = X )
            & ( distinct_real @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_6699_distinct__finite__set,axiom,
    ! [X: set_o] :
      ( finite_finite_list_o
      @ ( collect_list_o
        @ ^ [Ys3: list_o] :
            ( ( ( set_o2 @ Ys3 )
              = X )
            & ( distinct_o @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_6700_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P7: nat] : ( times_times_complex @ ( power_power_complex @ X @ P7 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P7 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6701_diff__power__eq__sum,axiom,
    ! [X: uint32,N: nat,Y: uint32] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ X @ ( suc @ N ) ) @ ( power_power_uint32 @ Y @ ( suc @ N ) ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ X @ Y )
        @ ( groups833757482993574392uint32
          @ ^ [P7: nat] : ( times_times_uint32 @ ( power_power_uint32 @ X @ P7 ) @ ( power_power_uint32 @ Y @ ( minus_minus_nat @ N @ P7 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6702_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P7: nat] : ( times_times_int @ ( power_power_int @ X @ P7 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P7 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6703_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P7: nat] : ( times_times_real @ ( power_power_real @ X @ P7 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P7 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6704_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6705_power__diff__sumr2,axiom,
    ! [X: uint32,N: nat,Y: uint32] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ X @ N ) @ ( power_power_uint32 @ Y @ N ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ X @ Y )
        @ ( groups833757482993574392uint32
          @ ^ [I3: nat] : ( times_times_uint32 @ ( power_power_uint32 @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_uint32 @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6706_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6707_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6708_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > real,B4: real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S )
            = B4 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6709_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B4: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B4 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6710_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B4: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B4 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6711_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B4: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B4 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6712_sum__nonneg__leq__bound,axiom,
    ! [S: set_Code_integer,F: code_integer > real,B4: real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S )
            = B4 )
         => ( ( member_Code_integer @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6713_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,B4: rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = B4 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6714_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B4: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B4 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6715_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B4: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B4 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6716_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B4: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B4 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6717_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B4: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B4 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6718_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S )
            = zero_zero_real )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6719_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6720_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6721_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6722_sum__nonneg__0,axiom,
    ! [S: set_Code_integer,F: code_integer > real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S )
            = zero_zero_real )
         => ( ( member_Code_integer @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6723_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6724_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6725_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6726_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6727_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6728_add__mset__in__multiset,axiom,
    ! [M7: list_nat > nat,A: list_nat] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M7 @ X4 ) ) @ ( M7 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_6729_add__mset__in__multiset,axiom,
    ! [M7: set_nat > nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M7 @ X4 ) ) @ ( M7 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_6730_add__mset__in__multiset,axiom,
    ! [M7: nat > nat,A: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M7 @ X4 ) ) @ ( M7 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_6731_add__mset__in__multiset,axiom,
    ! [M7: int > nat,A: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M7 @ X4 ) ) @ ( M7 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_6732_add__mset__in__multiset,axiom,
    ! [M7: complex > nat,A: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M7 @ X4 ) ) @ ( M7 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_6733_add__mset__in__multiset,axiom,
    ! [M7: code_integer > nat,A: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M7 @ X4 ) ) @ ( M7 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_6734_sum__fun__comp,axiom,
    ! [S3: set_VEBT_VEBT,R2: set_nat,G: vEBT_VEBT > nat,F: nat > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_VEBT_VEBT_nat @ G @ S3 ) @ R2 )
         => ( ( groups8325533452322294502uint32
              @ ^ [X4: vEBT_VEBT] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups833757482993574392uint32
              @ ^ [Y4: nat] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite7802652506058667612T_VEBT
                      @ ( collect_VEBT_VEBT
                        @ ^ [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6735_sum__fun__comp,axiom,
    ! [S3: set_real,R2: set_nat,G: real > nat,F: nat > uint32] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_real_nat @ G @ S3 ) @ R2 )
         => ( ( groups5944083974425963860uint32
              @ ^ [X4: real] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups833757482993574392uint32
              @ ^ [Y4: nat] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite_card_real
                      @ ( collect_real
                        @ ^ [X4: real] :
                            ( ( member_real @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6736_sum__fun__comp,axiom,
    ! [S3: set_literal,R2: set_nat,G: literal > nat,F: nat > uint32] :
      ( ( finite5847741373460823677iteral @ S3 )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_literal_nat @ G @ S3 ) @ R2 )
         => ( ( groups4542640294765070598uint32
              @ ^ [X4: literal] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups833757482993574392uint32
              @ ^ [Y4: nat] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite_card_literal
                      @ ( collect_literal
                        @ ^ [X4: literal] :
                            ( ( member_literal @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6737_sum__fun__comp,axiom,
    ! [S3: set_VEBT_VEBT,R2: set_complex,G: vEBT_VEBT > complex,F: complex > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( finite3207457112153483333omplex @ R2 )
       => ( ( ord_le211207098394363844omplex @ ( image_3793382806556112285omplex @ G @ S3 ) @ R2 )
         => ( ( groups8325533452322294502uint32
              @ ^ [X4: vEBT_VEBT] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups8736914816313324502uint32
              @ ^ [Y4: complex] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite7802652506058667612T_VEBT
                      @ ( collect_VEBT_VEBT
                        @ ^ [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6738_sum__fun__comp,axiom,
    ! [S3: set_real,R2: set_complex,G: real > complex,F: complex > uint32] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite3207457112153483333omplex @ R2 )
       => ( ( ord_le211207098394363844omplex @ ( image_real_complex @ G @ S3 ) @ R2 )
         => ( ( groups5944083974425963860uint32
              @ ^ [X4: real] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups8736914816313324502uint32
              @ ^ [Y4: complex] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite_card_real
                      @ ( collect_real
                        @ ^ [X4: real] :
                            ( ( member_real @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6739_sum__fun__comp,axiom,
    ! [S3: set_literal,R2: set_complex,G: literal > complex,F: complex > uint32] :
      ( ( finite5847741373460823677iteral @ S3 )
     => ( ( finite3207457112153483333omplex @ R2 )
       => ( ( ord_le211207098394363844omplex @ ( image_5274195009022015549omplex @ G @ S3 ) @ R2 )
         => ( ( groups4542640294765070598uint32
              @ ^ [X4: literal] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups8736914816313324502uint32
              @ ^ [Y4: complex] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite_card_literal
                      @ ( collect_literal
                        @ ^ [X4: literal] :
                            ( ( member_literal @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6740_sum__fun__comp,axiom,
    ! [S3: set_VEBT_VEBT,R2: set_Code_integer,G: vEBT_VEBT > code_integer,F: code_integer > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( finite6017078050557962740nteger @ R2 )
       => ( ( ord_le7084787975880047091nteger @ ( image_2092689629700589388nteger @ G @ S3 ) @ R2 )
         => ( ( groups8325533452322294502uint32
              @ ^ [X4: vEBT_VEBT] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups8847630953604152069uint32
              @ ^ [Y4: code_integer] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite7802652506058667612T_VEBT
                      @ ( collect_VEBT_VEBT
                        @ ^ [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6741_sum__fun__comp,axiom,
    ! [S3: set_real,R2: set_Code_integer,G: real > code_integer,F: code_integer > uint32] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite6017078050557962740nteger @ R2 )
       => ( ( ord_le7084787975880047091nteger @ ( image_4958697645175560720nteger @ G @ S3 ) @ R2 )
         => ( ( groups5944083974425963860uint32
              @ ^ [X4: real] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups8847630953604152069uint32
              @ ^ [Y4: code_integer] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite_card_real
                      @ ( collect_real
                        @ ^ [X4: real] :
                            ( ( member_real @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6742_sum__fun__comp,axiom,
    ! [S3: set_literal,R2: set_Code_integer,G: literal > code_integer,F: code_integer > uint32] :
      ( ( finite5847741373460823677iteral @ S3 )
     => ( ( finite6017078050557962740nteger @ R2 )
       => ( ( ord_le7084787975880047091nteger @ ( image_4442872163159017964nteger @ G @ S3 ) @ R2 )
         => ( ( groups4542640294765070598uint32
              @ ^ [X4: literal] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups8847630953604152069uint32
              @ ^ [Y4: code_integer] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite_card_literal
                      @ ( collect_literal
                        @ ^ [X4: literal] :
                            ( ( member_literal @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6743_sum__fun__comp,axiom,
    ! [S3: set_nat,R2: set_nat,G: nat > nat,F: nat > uint32] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ S3 ) @ R2 )
         => ( ( groups833757482993574392uint32
              @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
              @ S3 )
            = ( groups833757482993574392uint32
              @ ^ [Y4: nat] :
                  ( times_times_uint32
                  @ ( semiri2565882477558803405uint32
                    @ ( finite_card_nat
                      @ ( collect_nat
                        @ ^ [X4: nat] :
                            ( ( member_nat @ X4 @ S3 )
                            & ( ( G @ X4 )
                              = Y4 ) ) ) ) )
                  @ ( F @ Y4 ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_6744_diff__preserves__multiset,axiom,
    ! [M7: list_nat > nat,N8: list_nat > nat] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M7 @ X4 ) @ ( N8 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_6745_diff__preserves__multiset,axiom,
    ! [M7: set_nat > nat,N8: set_nat > nat] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M7 @ X4 ) @ ( N8 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_6746_diff__preserves__multiset,axiom,
    ! [M7: nat > nat,N8: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M7 @ X4 ) @ ( N8 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_6747_diff__preserves__multiset,axiom,
    ! [M7: int > nat,N8: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M7 @ X4 ) @ ( N8 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_6748_diff__preserves__multiset,axiom,
    ! [M7: complex > nat,N8: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M7 @ X4 ) @ ( N8 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_6749_diff__preserves__multiset,axiom,
    ! [M7: code_integer > nat,N8: code_integer > nat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M7 @ X4 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M7 @ X4 ) @ ( N8 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_6750_sum_Ogroup,axiom,
    ! [S3: set_nat,T3: set_complex,G: nat > complex,H2: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite3207457112153483333omplex @ T3 )
       => ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ G @ S3 ) @ T3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [Y4: complex] :
                  ( groups3542108847815614940at_nat @ H2
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups3542108847815614940at_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6751_sum_Ogroup,axiom,
    ! [S3: set_nat,T3: set_Code_integer,G: nat > code_integer,H2: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite6017078050557962740nteger @ T3 )
       => ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ G @ S3 ) @ T3 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [Y4: code_integer] :
                  ( groups3542108847815614940at_nat @ H2
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups3542108847815614940at_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6752_sum_Ogroup,axiom,
    ! [S3: set_nat,T3: set_int,G: nat > int,H2: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite_finite_int @ T3 )
       => ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S3 ) @ T3 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [Y4: int] :
                  ( groups3542108847815614940at_nat @ H2
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups3542108847815614940at_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6753_sum_Ogroup,axiom,
    ! [S3: set_complex,T3: set_nat,G: complex > nat,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite_finite_nat @ T3 )
       => ( ( ord_less_eq_set_nat @ ( image_complex_nat @ G @ S3 ) @ T3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [Y4: nat] :
                  ( groups7754918857620584856omplex @ H2
                  @ ( collect_complex
                    @ ^ [X4: complex] :
                        ( ( member_complex @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups7754918857620584856omplex @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6754_sum_Ogroup,axiom,
    ! [S3: set_complex,T3: set_Code_integer,G: complex > code_integer,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite6017078050557962740nteger @ T3 )
       => ( ( ord_le7084787975880047091nteger @ ( image_1994230757181692690nteger @ G @ S3 ) @ T3 )
         => ( ( groups8024822376189712711omplex
              @ ^ [Y4: code_integer] :
                  ( groups7754918857620584856omplex @ H2
                  @ ( collect_complex
                    @ ^ [X4: complex] :
                        ( ( member_complex @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups7754918857620584856omplex @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6755_sum_Ogroup,axiom,
    ! [S3: set_complex,T3: set_int,G: complex > int,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite_finite_int @ T3 )
       => ( ( ord_less_eq_set_int @ ( image_complex_int @ G @ S3 ) @ T3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [Y4: int] :
                  ( groups7754918857620584856omplex @ H2
                  @ ( collect_complex
                    @ ^ [X4: complex] :
                        ( ( member_complex @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups7754918857620584856omplex @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6756_sum_Ogroup,axiom,
    ! [S3: set_nat,T3: set_complex,G: nat > complex,H2: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite3207457112153483333omplex @ T3 )
       => ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ G @ S3 ) @ T3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [Y4: complex] :
                  ( groups6591440286371151544t_real @ H2
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups6591440286371151544t_real @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6757_sum_Ogroup,axiom,
    ! [S3: set_nat,T3: set_Code_integer,G: nat > code_integer,H2: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite6017078050557962740nteger @ T3 )
       => ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ G @ S3 ) @ T3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [Y4: code_integer] :
                  ( groups6591440286371151544t_real @ H2
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups6591440286371151544t_real @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6758_sum_Ogroup,axiom,
    ! [S3: set_nat,T3: set_int,G: nat > int,H2: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite_finite_int @ T3 )
       => ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S3 ) @ T3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [Y4: int] :
                  ( groups6591440286371151544t_real @ H2
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups6591440286371151544t_real @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6759_sum_Ogroup,axiom,
    ! [S3: set_int,T3: set_nat,G: int > nat,H2: int > int] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite_finite_nat @ T3 )
       => ( ( ord_less_eq_set_nat @ ( image_int_nat @ G @ S3 ) @ T3 )
         => ( ( groups3539618377306564664at_int
              @ ^ [Y4: nat] :
                  ( groups4538972089207619220nt_int @ H2
                  @ ( collect_int
                    @ ^ [X4: int] :
                        ( ( member_int @ X4 @ S3 )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T3 )
            = ( groups4538972089207619220nt_int @ H2 @ S3 ) ) ) ) ) ).

% sum.group
thf(fact_6760_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > uint32] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups5712668689793887828uint32 @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_uint32 ) ) ) )
        = ( groups5712668689793887828uint32 @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6761_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups8736914816313324502uint32 @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_uint32 ) ) ) )
        = ( groups8736914816313324502uint32 @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6762_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups8847630953604152069uint32 @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_uint32 ) ) ) )
        = ( groups8847630953604152069uint32 @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6763_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6764_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6765_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1270011288395367621r_real @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups1270011288395367621r_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6766_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6767_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6768_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6769_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6770_sum__Suc,axiom,
    ! [F: complex > nat,A4: set_complex] :
      ( ( groups5693394587270226106ex_nat
        @ ^ [X4: complex] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( finite_card_complex @ A4 ) ) ) ).

% sum_Suc
thf(fact_6771_sum__Suc,axiom,
    ! [F: literal > nat,A4: set_literal] :
      ( ( groups8652099787943017962al_nat
        @ ^ [X4: literal] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups8652099787943017962al_nat @ F @ A4 ) @ ( finite_card_literal @ A4 ) ) ) ).

% sum_Suc
thf(fact_6772_sum__Suc,axiom,
    ! [F: list_nat > nat,A4: set_list_nat] :
      ( ( groups4396056296759096172at_nat
        @ ^ [X4: list_nat] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups4396056296759096172at_nat @ F @ A4 ) @ ( finite_card_list_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_6773_sum__Suc,axiom,
    ! [F: set_nat > nat,A4: set_set_nat] :
      ( ( groups8294997508430121362at_nat
        @ ^ [X4: set_nat] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( finite_card_set_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_6774_sum__Suc,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( finite_card_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_6775_sum__multicount,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,R2: vEBT_VEBT > vEBT_VEBT > $o,K: nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( finite5795047828879050333T_VEBT @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ T3 )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [J3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite7802652506058667612T_VEBT @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6776_sum__multicount,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_real,R2: vEBT_VEBT > real > $o,K: nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( finite_finite_real @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T3 )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite_card_real @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6777_sum__multicount,axiom,
    ! [S3: set_real,T3: set_VEBT_VEBT,R2: real > vEBT_VEBT > $o,K: nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite5795047828879050333T_VEBT @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ T3 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I3: real] :
                        ( ( member_real @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I3: real] :
                  ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [J3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite7802652506058667612T_VEBT @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6778_sum__multicount,axiom,
    ! [S3: set_real,T3: set_real,R2: real > real > $o,K: nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite_finite_real @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T3 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I3: real] :
                        ( ( member_real @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I3: real] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite_card_real @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6779_sum__multicount,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_literal,R2: vEBT_VEBT > literal > $o,K: nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( finite5847741373460823677iteral @ T3 )
       => ( ! [X3: literal] :
              ( ( member_literal @ X3 @ T3 )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [J3: literal] :
                        ( ( member_literal @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite_card_literal @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6780_sum__multicount,axiom,
    ! [S3: set_real,T3: set_literal,R2: real > literal > $o,K: nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite5847741373460823677iteral @ T3 )
       => ( ! [X3: literal] :
              ( ( member_literal @ X3 @ T3 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I3: real] :
                        ( ( member_real @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I3: real] :
                  ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [J3: literal] :
                        ( ( member_literal @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite_card_literal @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6781_sum__multicount,axiom,
    ! [S3: set_literal,T3: set_VEBT_VEBT,R2: literal > vEBT_VEBT > $o,K: nat] :
      ( ( finite5847741373460823677iteral @ S3 )
     => ( ( finite5795047828879050333T_VEBT @ T3 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ T3 )
             => ( ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [I3: literal] :
                        ( ( member_literal @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups8652099787943017962al_nat
              @ ^ [I3: literal] :
                  ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [J3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite7802652506058667612T_VEBT @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6782_sum__multicount,axiom,
    ! [S3: set_literal,T3: set_real,R2: literal > real > $o,K: nat] :
      ( ( finite5847741373460823677iteral @ S3 )
     => ( ( finite_finite_real @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T3 )
             => ( ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [I3: literal] :
                        ( ( member_literal @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups8652099787943017962al_nat
              @ ^ [I3: literal] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite_card_real @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6783_sum__multicount,axiom,
    ! [S3: set_literal,T3: set_literal,R2: literal > literal > $o,K: nat] :
      ( ( finite5847741373460823677iteral @ S3 )
     => ( ( finite5847741373460823677iteral @ T3 )
       => ( ! [X3: literal] :
              ( ( member_literal @ X3 @ T3 )
             => ( ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [I3: literal] :
                        ( ( member_literal @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups8652099787943017962al_nat
              @ ^ [I3: literal] :
                  ( finite_card_literal
                  @ ( collect_literal
                    @ ^ [J3: literal] :
                        ( ( member_literal @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite_card_literal @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6784_sum__multicount,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_nat,R2: vEBT_VEBT > nat > $o,K: nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( finite_finite_nat @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T3 )
             => ( ( finite7802652506058667612T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [I3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ I3 @ S3 )
                        & ( R2 @ I3 @ X3 ) ) ) )
                = K ) )
         => ( ( groups771621172384141258BT_nat
              @ ^ [I3: vEBT_VEBT] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J3: nat] :
                        ( ( member_nat @ J3 @ T3 )
                        & ( R2 @ I3 @ J3 ) ) ) )
              @ S3 )
            = ( times_times_nat @ K @ ( finite_card_nat @ T3 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6785_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_6786_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_6787_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_6788_foldr__length__aux,axiom,
    ! [L: list_VEBT_VEBT,A: nat] :
      ( ( foldr_VEBT_VEBT_nat
        @ ^ [X4: vEBT_VEBT] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_s6755466524823107622T_VEBT @ L ) ) ) ).

% foldr_length_aux
thf(fact_6789_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] :
      ( ( foldr_real_nat
        @ ^ [X4: real] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_real @ L ) ) ) ).

% foldr_length_aux
thf(fact_6790_foldr__length__aux,axiom,
    ! [L: list_o,A: nat] :
      ( ( foldr_o_nat
        @ ^ [X4: $o] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_o @ L ) ) ) ).

% foldr_length_aux
thf(fact_6791_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] :
      ( ( foldr_nat_nat
        @ ^ [X4: nat] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_nat @ L ) ) ) ).

% foldr_length_aux
thf(fact_6792_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_nat
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ X4 )
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_set_nat @ Y4 @ X4 )
    @ bot_bot_set_nat ) ).

% bot.ordering_top_axioms
thf(fact_6793_bot_Oordering__top__axioms,axiom,
    ( ordering_top_assn
    @ ^ [X4: assn,Y4: assn] : ( ord_less_eq_assn @ Y4 @ X4 )
    @ ^ [X4: assn,Y4: assn] : ( ord_less_assn @ Y4 @ X4 )
    @ bot_bot_assn ) ).

% bot.ordering_top_axioms
thf(fact_6794_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_o
    @ ^ [X4: set_o,Y4: set_o] : ( ord_less_eq_set_o @ Y4 @ X4 )
    @ ^ [X4: set_o,Y4: set_o] : ( ord_less_set_o @ Y4 @ X4 )
    @ bot_bot_set_o ) ).

% bot.ordering_top_axioms
thf(fact_6795_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_int
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ X4 )
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_set_int @ Y4 @ X4 )
    @ bot_bot_set_int ) ).

% bot.ordering_top_axioms
thf(fact_6796_bot_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
    @ bot_bot_nat ) ).

% bot.ordering_top_axioms
thf(fact_6797_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_6798_set__zip__rightD,axiom,
    ! [X: $o,Y: $o,Xs: list_o,Ys: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) )
     => ( member_o @ Y @ ( set_o2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_6799_set__zip__rightD,axiom,
    ! [X: uint32,Y: uint32,Xs: list_uint32,Ys: list_uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ ( set_Pr2418681094996576974uint32 @ ( zip_uint32_uint32 @ Xs @ Ys ) ) )
     => ( member_uint32 @ Y @ ( set_uint322 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_6800_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_6801_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_6802_set__zip__rightD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ ( set_Pr364071364088684201et_nat @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) ) )
     => ( member6260224972018164377et_nat @ Y @ ( set_Pr3864771053587467565et_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_6803_set__zip__rightD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P2321686559999237006et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ ( set_Pr7458301993993767461et_nat @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) ) )
     => ( member1996754912294343701et_nat @ Y @ ( set_Pr364071364088684201et_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_6804_set__zip__leftD,axiom,
    ! [X: $o,Y: $o,Xs: list_o,Ys: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) )
     => ( member_o @ X @ ( set_o2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_6805_set__zip__leftD,axiom,
    ! [X: uint32,Y: uint32,Xs: list_uint32,Ys: list_uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ ( set_Pr2418681094996576974uint32 @ ( zip_uint32_uint32 @ Xs @ Ys ) ) )
     => ( member_uint32 @ X @ ( set_uint322 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_6806_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_6807_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int @ X @ ( set_int2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_6808_set__zip__leftD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ ( set_Pr364071364088684201et_nat @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) ) )
     => ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_6809_set__zip__leftD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P2321686559999237006et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ ( set_Pr7458301993993767461et_nat @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) ) )
     => ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_6810_in__set__zipE,axiom,
    ! [X: int,Y: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X @ Y ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6811_in__set__zipE,axiom,
    ! [X: int,Y: nat,Xs: list_int,Ys: list_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ ( set_Pr6647972299459129970nt_nat @ ( zip_int_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6812_in__set__zipE,axiom,
    ! [X: int,Y: real,Xs: list_int,Ys: list_real] :
      ( ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X @ Y ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6813_in__set__zipE,axiom,
    ! [X: int,Y: $o,Xs: list_int,Ys: list_o] :
      ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ X @ Y ) @ ( set_Pr8694291782656941196_int_o @ ( zip_int_o @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_o @ Y @ ( set_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6814_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: int,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6815_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6816_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6817_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: real,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6818_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: $o,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X @ Y ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_o @ Y @ ( set_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6819_in__set__zipE,axiom,
    ! [X: nat,Y: int,Xs: list_nat,Ys: list_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y ) @ ( set_Pr1470767568048878706at_int @ ( zip_nat_int @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_6820_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
      = ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_6821_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
      = ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_6822_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
      = ( ( member_real @ A @ ( set_real2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_6823_zip__same,axiom,
    ! [A: $o,B: $o,Xs: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Xs ) ) )
      = ( ( member_o @ A @ ( set_o2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_6824_zip__same,axiom,
    ! [A: uint32,B: uint32,Xs: list_uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ A @ B ) @ ( set_Pr2418681094996576974uint32 @ ( zip_uint32_uint32 @ Xs @ Xs ) ) )
      = ( ( member_uint32 @ A @ ( set_uint322 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_6825_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_6826_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
      = ( ( member_int @ A @ ( set_int2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_6827_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6828_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6829_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6830_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6831_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6832_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_6833_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_6834_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_6835_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_6836_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_6837_zip__update,axiom,
    ! [Xs: list_o,I: nat,X: $o,Ys: list_o,Y: $o] :
      ( ( zip_o_o @ ( list_update_o @ Xs @ I @ X ) @ ( list_update_o @ Ys @ I @ Y ) )
      = ( list_u1537252308907898773od_o_o @ ( zip_o_o @ Xs @ Ys ) @ I @ ( product_Pair_o_o @ X @ Y ) ) ) ).

% zip_update
thf(fact_6838_zip__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( zip_VE793581609497812771_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Ys @ I @ Y ) )
      = ( list_u30558089781959097_VEBTi @ ( zip_VE793581609497812771_VEBTi @ Xs @ Ys ) @ I @ ( produc436343169921013763_VEBTi @ X @ Y ) ) ) ).

% zip_update
thf(fact_6839_zip__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( zip_VE7413257051550508102T_VEBT @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
      = ( list_u9044509791230035014T_VEBT @ ( zip_VE7413257051550508102T_VEBT @ Xs @ Ys ) @ I @ ( produc7053807326796202854T_VEBT @ X @ Y ) ) ) ).

% zip_update
thf(fact_6840_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( zip_VE6444338338598820466_VEBTi @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Ys @ I @ Y ) )
      = ( list_u636002643828906794_VEBTi @ ( zip_VE6444338338598820466_VEBTi @ Xs @ Ys ) @ I @ ( produc6084888613844515218_VEBTi @ X @ Y ) ) ) ).

% zip_update
thf(fact_6841_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( zip_VE537291747668921783T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
      = ( list_u6961636818849549845T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I @ ( produc537772716801021591T_VEBT @ X @ Y ) ) ) ).

% zip_update
thf(fact_6842_zip__update,axiom,
    ! [Xs: list_uint32,I: nat,X: uint32,Ys: list_uint32,Y: uint32] :
      ( ( zip_uint32_uint32 @ ( list_update_uint32 @ Xs @ I @ X ) @ ( list_update_uint32 @ Ys @ I @ Y ) )
      = ( list_u2546161786664092711uint32 @ ( zip_uint32_uint32 @ Xs @ Ys ) @ I @ ( produc1400373151660368625uint32 @ X @ Y ) ) ) ).

% zip_update
thf(fact_6843_zip__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( zip_nat_nat @ ( list_update_nat @ Xs @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) )
      = ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_6844_zip__update,axiom,
    ! [Xs: list_int,I: nat,X: int,Ys: list_int,Y: int] :
      ( ( zip_int_int @ ( list_update_int @ Xs @ I @ X ) @ ( list_update_int @ Ys @ I @ Y ) )
      = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ ( product_Pair_int_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_6845_zip__update,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,I: nat,X: produc3658429121746597890et_nat > $o,Ys: list_P9062070895058802706et_nat,Y: produc3658429121746597890et_nat] :
      ( ( zip_Pr7134870689397686104et_nat @ ( list_u6943956310655620667_nat_o @ Xs @ I @ X ) @ ( list_u9060326803697358356et_nat @ Ys @ I @ Y ) )
      = ( list_u1866171116859985808et_nat @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) @ I @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_6846_zip__update,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,I: nat,X: produc3658429121746597890et_nat > $o,Ys: list_P2321686559999237006et_nat,Y: produc3925858234332021118et_nat] :
      ( ( zip_Pr8136144321567152340et_nat @ ( list_u6943956310655620667_nat_o @ Xs @ I @ X ) @ ( list_u1866171116859985808et_nat @ Ys @ I @ Y ) )
      = ( list_u1913845194572276492et_nat @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) @ I @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_6847_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6848_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6849_one__diff__power__eq_H,axiom,
    ! [X: uint32,N: nat] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( power_power_uint32 @ X @ N ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X )
        @ ( groups833757482993574392uint32
          @ ^ [I3: nat] : ( power_power_uint32 @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6850_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6851_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6852_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_6853_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_6854_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_6855_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_6856_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N ) )
     != zero_z3403309356797280102nteger ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_6857_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6858_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6859_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6860_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6861_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6862_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6863_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6864_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6865_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6866_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6867_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_6868_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_6869_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_6870_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_6871_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_6872_Iio__eq__empty__iff,axiom,
    ! [N: $o] :
      ( ( ( set_ord_lessThan_o @ N )
        = bot_bot_set_o )
      = ( N = bot_bot_o ) ) ).

% Iio_eq_empty_iff
thf(fact_6873_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_6874_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_6875_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
      = ( ord_less_rat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6876_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6877_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6878_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
      = ( ord_less_int @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6879_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
      = ( ord_less_real @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6880_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_6881_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_6882_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: nat > uint32] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups833757482993574392uint32
          @ ^ [I3: nat] : ( minus_minus_uint32 @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( minus_minus_uint32 @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_6883_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_6884_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_6885_Max__add__commute,axiom,
    ! [S3: set_complex,F: complex > real,K: real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic4275903605611617917x_real
            @ ( image_complex_real
              @ ^ [X4: complex] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_complex_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6886_Max__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > real,K: real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic4275903605611617917x_real
            @ ( image_7738145705984076560r_real
              @ ^ [X4: code_integer] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_7738145705984076560r_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6887_Max__add__commute,axiom,
    ! [S3: set_complex,F: complex > rat,K: rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic7630753665789217321ax_rat
            @ ( image_complex_rat
              @ ^ [X4: complex] : ( plus_plus_rat @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_complex_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6888_Max__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > rat,K: rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic7630753665789217321ax_rat
            @ ( image_315895873841295420er_rat
              @ ^ [X4: code_integer] : ( plus_plus_rat @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_315895873841295420er_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6889_Max__add__commute,axiom,
    ! [S3: set_complex,F: complex > int,K: int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_complex_int
              @ ^ [X4: complex] : ( plus_plus_int @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_int @ ( lattic8263393255366662781ax_int @ ( image_complex_int @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6890_Max__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > int,K: int] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_948535463418740880er_int
              @ ^ [X4: code_integer] : ( plus_plus_int @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_int @ ( lattic8263393255366662781ax_int @ ( image_948535463418740880er_int @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6891_Max__add__commute,axiom,
    ! [S3: set_nat,F: nat > real,K: real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( lattic4275903605611617917x_real
            @ ( image_nat_real
              @ ^ [X4: nat] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_nat_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6892_Max__add__commute,axiom,
    ! [S3: set_nat,F: nat > rat,K: rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( lattic7630753665789217321ax_rat
            @ ( image_nat_rat
              @ ^ [X4: nat] : ( plus_plus_rat @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_nat_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6893_Max__add__commute,axiom,
    ! [S3: set_nat,F: nat > int,K: int] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_nat_int
              @ ^ [X4: nat] : ( plus_plus_int @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_int @ ( lattic8263393255366662781ax_int @ ( image_nat_int @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6894_Max__add__commute,axiom,
    ! [S3: set_int,F: int > real,K: real] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( lattic4275903605611617917x_real
            @ ( image_int_real
              @ ^ [X4: int] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_int_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6895_Min__add__commute,axiom,
    ! [S3: set_complex,F: complex > real,K: real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic3629708407755379051n_real
            @ ( image_complex_real
              @ ^ [X4: complex] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic3629708407755379051n_real @ ( image_complex_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6896_Min__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > real,K: real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic3629708407755379051n_real
            @ ( image_7738145705984076560r_real
              @ ^ [X4: code_integer] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic3629708407755379051n_real @ ( image_7738145705984076560r_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6897_Min__add__commute,axiom,
    ! [S3: set_complex,F: complex > rat,K: rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic8086005427650270231in_rat
            @ ( image_complex_rat
              @ ^ [X4: complex] : ( plus_plus_rat @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic8086005427650270231in_rat @ ( image_complex_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6898_Min__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > rat,K: rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic8086005427650270231in_rat
            @ ( image_315895873841295420er_rat
              @ ^ [X4: code_integer] : ( plus_plus_rat @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic8086005427650270231in_rat @ ( image_315895873841295420er_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6899_Min__add__commute,axiom,
    ! [S3: set_complex,F: complex > int,K: int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic8718645017227715691in_int
            @ ( image_complex_int
              @ ^ [X4: complex] : ( plus_plus_int @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_int @ ( lattic8718645017227715691in_int @ ( image_complex_int @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6900_Min__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > int,K: int] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic8718645017227715691in_int
            @ ( image_948535463418740880er_int
              @ ^ [X4: code_integer] : ( plus_plus_int @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_int @ ( lattic8718645017227715691in_int @ ( image_948535463418740880er_int @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6901_Min__add__commute,axiom,
    ! [S3: set_nat,F: nat > real,K: real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( lattic3629708407755379051n_real
            @ ( image_nat_real
              @ ^ [X4: nat] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic3629708407755379051n_real @ ( image_nat_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6902_Min__add__commute,axiom,
    ! [S3: set_nat,F: nat > rat,K: rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( lattic8086005427650270231in_rat
            @ ( image_nat_rat
              @ ^ [X4: nat] : ( plus_plus_rat @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic8086005427650270231in_rat @ ( image_nat_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6903_Min__add__commute,axiom,
    ! [S3: set_nat,F: nat > int,K: int] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( lattic8718645017227715691in_int
            @ ( image_nat_int
              @ ^ [X4: nat] : ( plus_plus_int @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_int @ ( lattic8718645017227715691in_int @ ( image_nat_int @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6904_Min__add__commute,axiom,
    ! [S3: set_int,F: int > real,K: real] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( lattic3629708407755379051n_real
            @ ( image_int_real
              @ ^ [X4: int] : ( plus_plus_real @ ( F @ X4 ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic3629708407755379051n_real @ ( image_int_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6905_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I3 ) ) )
        @ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_6906_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I3 ) ) )
        @ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_6907_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_6908_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_6909_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_6910_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y ) ) ) ).

% of_nat_max
thf(fact_6911_finite__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
              & ( ( size_s3451745648224563538omplex @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6912_finite__lists__length__eq,axiom,
    ! [A4: set_Code_integer,N: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs2: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs2 ) @ A4 )
              & ( ( size_s3445333598471063425nteger @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6913_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6914_finite__lists__length__eq,axiom,
    ! [A4: set_real,N: nat] :
      ( ( finite_finite_real @ A4 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs2: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 )
              & ( ( size_size_list_real @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6915_finite__lists__length__eq,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A4 )
              & ( ( size_size_list_o @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6916_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
              & ( ( size_size_list_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6917_finite__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
              & ( ( size_size_list_int @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6918_distinct__finite__subset,axiom,
    ! [X: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ X )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Ys3: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Ys3 ) @ X )
              & ( distinct_VEBT_VEBT @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_6919_distinct__finite__subset,axiom,
    ! [X: set_real] :
      ( ( finite_finite_real @ X )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Ys3: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Ys3 ) @ X )
              & ( distinct_real @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_6920_distinct__finite__subset,axiom,
    ! [X: set_o] :
      ( ( finite_finite_o @ X )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Ys3: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Ys3 ) @ X )
              & ( distinct_o @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_6921_distinct__finite__subset,axiom,
    ! [X: set_nat] :
      ( ( finite_finite_nat @ X )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Ys3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ X )
              & ( distinct_nat @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_6922_distinct__finite__subset,axiom,
    ! [X: set_complex] :
      ( ( finite3207457112153483333omplex @ X )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Ys3: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Ys3 ) @ X )
              & ( distinct_complex @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_6923_distinct__finite__subset,axiom,
    ! [X: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ X )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Ys3: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Ys3 ) @ X )
              & ( distin1543349897113766820nteger @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_6924_distinct__finite__subset,axiom,
    ! [X: set_int] :
      ( ( finite_finite_int @ X )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Ys3: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Ys3 ) @ X )
              & ( distinct_int @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_6925_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [K2: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% finite_nat_bounded
thf(fact_6926_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_6927_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real,C2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups2240296850493347238T_real @ C2 @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups2240296850493347238T_real @ C2 @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6928_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > real,C2: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups8097168146408367636l_real @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6929_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C2: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups5808333547571424918x_real @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6930_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > real,C2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups1270011288395367621r_real @ C2 @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups1270011288395367621r_real @ C2 @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6931_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat,C2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups136491112297645522BT_rat @ C2 @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups136491112297645522BT_rat @ C2 @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6932_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > rat,C2: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups1300246762558778688al_rat @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6933_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C2: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups5058264527183730370ex_rat @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6934_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > rat,C2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups6602215022474089585er_rat @ C2 @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups6602215022474089585er_rat @ C2 @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6935_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > nat,C2: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K3: vEBT_VEBT] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups771621172384141258BT_nat @ C2 @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K3: vEBT_VEBT] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups771621172384141258BT_nat @ C2 @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6936_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > nat,C2: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups1935376822645274424al_nat @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6937_card__lists__length__eq,axiom,
    ! [A4: set_literal,N: nat] :
      ( ( finite5847741373460823677iteral @ A4 )
     => ( ( finite7554361855635516162iteral
          @ ( collect_list_literal
            @ ^ [Xs2: list_literal] :
                ( ( ord_le7307670543136651348iteral @ ( set_literal2 @ Xs2 ) @ A4 )
                & ( ( size_s2501651207091587910iteral @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_literal @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6938_card__lists__length__eq,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite7325466520557071688st_nat
          @ ( collec5989764272469232197st_nat
            @ ^ [Xs2: list_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A4 )
                & ( ( size_s3023201423986296836st_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_list_nat @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6939_card__lists__length__eq,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite5631907774883551598et_nat
          @ ( collect_list_set_nat
            @ ^ [Xs2: list_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A4 )
                & ( ( size_s3254054031482475050et_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_set_nat @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6940_card__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite5120063068150530198omplex
          @ ( collect_list_complex
            @ ^ [Xs2: list_complex] :
                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
                & ( ( size_s3451745648224563538omplex @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_complex @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6941_card__lists__length__eq,axiom,
    ! [A4: set_Code_integer,N: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite5823187341872139973nteger
          @ ( collec3483841146883906114nteger
            @ ^ [Xs2: list_Code_integer] :
                ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs2 ) @ A4 )
                & ( ( size_s3445333598471063425nteger @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite4902975817058060853nteger @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6942_card__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite5915292604075114978T_VEBT
          @ ( collec5608196760682091941T_VEBT
            @ ^ [Xs2: list_VEBT_VEBT] :
                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6943_card__lists__length__eq,axiom,
    ! [A4: set_real,N: nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite4141703471298347796t_real
          @ ( collect_list_real
            @ ^ [Xs2: list_real] :
                ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 )
                & ( ( size_size_list_real @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_real @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6944_card__lists__length__eq,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_card_list_o
          @ ( collect_list_o
            @ ^ [Xs2: list_o] :
                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A4 )
                & ( ( size_size_list_o @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_o @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6945_card__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [Xs2: list_nat] :
                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
                & ( ( size_size_list_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_nat @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6946_card__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_card_list_int
          @ ( collect_list_int
            @ ^ [Xs2: list_int] :
                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
                & ( ( size_size_list_int @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_int @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_6947_finite__lists__length__le,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6948_finite__lists__length__le,axiom,
    ! [A4: set_Code_integer,N: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs2: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs2 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6949_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6950_finite__lists__length__le,axiom,
    ! [A4: set_real,N: nat] :
      ( ( finite_finite_real @ A4 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs2: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6951_finite__lists__length__le,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6952_finite__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6953_finite__lists__length__le,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6954_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,Y: uint32] :
      ( ( ( size_s4844771616002835472uint32 @ Xs )
        = ( size_s4844771616002835472uint32 @ Ys ) )
     => ( ( member_uint32 @ Y @ ( set_uint322 @ Ys ) )
       => ~ ! [X3: uint32] :
              ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ Y ) @ ( set_Pr2418681094996576974uint32 @ ( zip_uint32_uint32 @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6955_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_int,Y: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: int] :
              ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6956_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6957_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6958_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X3 @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6959_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,Y: $o] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X3 @ Y ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6960_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6961_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_int,Y: int] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: real] :
              ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6962_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X3: real] :
              ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6963_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_real,Y: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
       => ~ ! [X3: real] :
              ~ ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X3 @ Y ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6964_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,X: uint32] :
      ( ( ( size_s4844771616002835472uint32 @ Xs )
        = ( size_s4844771616002835472uint32 @ Ys ) )
     => ( ( member_uint32 @ X @ ( set_uint322 @ Xs ) )
       => ~ ! [Y3: uint32] :
              ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y3 ) @ ( set_Pr2418681094996576974uint32 @ ( zip_uint32_uint32 @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6965_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_int,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: int] :
              ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6966_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X @ Y3 ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6967_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_real,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: real] :
              ~ ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X @ Y3 ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6968_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_o,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: $o] :
              ~ ( member4489920277610959864_int_o @ ( product_Pair_int_o @ X @ Y3 ) @ ( set_Pr8694291782656941196_int_o @ ( zip_int_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6969_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_nat,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y3 ) @ ( set_Pr6647972299459129970nt_nat @ ( zip_int_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6970_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6971_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: real] :
              ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6972_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: $o] :
              ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X @ Y3 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6973_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6974_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_6975_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_6976_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_6977_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_6978_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_6979_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_6980_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_6981_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri4939895301339042750nteger @ K )
       != zero_z3403309356797280102nteger )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_6982_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri2565882477558803405uint32 @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_uint32 @ ( semiri2565882477558803405uint32 @ M ) @ ( semiri2565882477558803405uint32 @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6983_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6984_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6985_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6986_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N ) )
        = ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6987_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( ( zip_int_nat @ Xs @ Ys )
        = ( cons_P7512249878480867347nt_nat @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs5: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs5 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6988_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_o,Xy: product_prod_int_o,Xys: list_P5087981734274514673_int_o] :
      ( ( ( zip_int_o @ Xs @ Ys )
        = ( cons_P7321330006258091179_int_o @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs5: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs5 ) )
           => ! [Y3: $o,Ys4: list_o] :
                ( ( Ys
                  = ( cons_o @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_o @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_o @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6989_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( ( zip_nat_int @ Xs @ Ys )
        = ( cons_P2335045147070616083at_int @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs5 ) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6990_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_o,Xy: product_prod_nat_o,Xys: list_P7333126701944960589_nat_o] :
      ( ( ( zip_nat_o @ Xs @ Ys )
        = ( cons_P9142372351690779143_nat_o @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs5 ) )
           => ! [Y3: $o,Ys4: list_o] :
                ( ( Ys
                  = ( cons_o @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_o @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_o @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6991_zip__eq__ConsE,axiom,
    ! [Xs: list_o,Ys: list_int,Xy: product_prod_o_int,Xys: list_P3795440434834930179_o_int] :
      ( ( ( zip_o_int @ Xs @ Ys )
        = ( cons_P1455986808126089405_o_int @ Xy @ Xys ) )
     => ~ ! [X3: $o,Xs5: list_o] :
            ( ( Xs
              = ( cons_o @ X3 @ Xs5 ) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_o_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_o_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6992_zip__eq__ConsE,axiom,
    ! [Xs: list_o,Ys: list_nat,Xy: product_prod_o_nat,Xys: list_P6285523579766656935_o_nat] :
      ( ( ( zip_o_nat @ Xs @ Ys )
        = ( cons_P5633837827635286113_o_nat @ Xy @ Xys ) )
     => ~ ! [X3: $o,Xs5: list_o] :
            ( ( Xs
              = ( cons_o @ X3 @ Xs5 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_o_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_o_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6993_zip__eq__ConsE,axiom,
    ! [Xs: list_o,Ys: list_o,Xy: product_prod_o_o,Xys: list_P4002435161011370285od_o_o] :
      ( ( ( zip_o_o @ Xs @ Ys )
        = ( cons_P8766293264717362397od_o_o @ Xy @ Xys ) )
     => ~ ! [X3: $o,Xs5: list_o] :
            ( ( Xs
              = ( cons_o @ X3 @ Xs5 ) )
           => ! [Y3: $o,Ys4: list_o] :
                ( ( Ys
                  = ( cons_o @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_o_o @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_o_o @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6994_zip__eq__ConsE,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,Xy: produc827990862158126777uint32,Xys: list_P3069071885182933823uint32] :
      ( ( ( zip_uint32_uint32 @ Xs @ Ys )
        = ( cons_P3149448846263281007uint32 @ Xy @ Xys ) )
     => ~ ! [X3: uint32,Xs5: list_uint32] :
            ( ( Xs
              = ( cons_uint32 @ X3 @ Xs5 ) )
           => ! [Y3: uint32,Ys4: list_uint32] :
                ( ( Ys
                  = ( cons_uint32 @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc1400373151660368625uint32 @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_uint32_uint32 @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6995_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs5 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6996_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( ( zip_int_int @ Xs @ Ys )
        = ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs5: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs5 ) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_6997_lenlex__irreflexive,axiom,
    ! [R: set_Pr1773385645901665561uint32,Xs: list_uint32] :
      ( ! [X3: uint32] :
          ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ X3 ) @ R )
     => ~ ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Xs @ Xs ) @ ( lenlex_uint32 @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_6998_lenlex__irreflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_6999_lenlex__irreflexive,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
     => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lenlex_int @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_7000_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y5: real] :
        ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_7001_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C2 @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C2 )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
      & ( ~ ( ord_less_nat @ C2 @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_7002_listrel1I1,axiom,
    ! [X: $o,Y: $o,R: set_Product_prod_o_o,Xs: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
     => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs ) @ ( cons_o @ Y @ Xs ) ) @ ( listrel1_o @ R ) ) ) ).

% listrel1I1
thf(fact_7003_listrel1I1,axiom,
    ! [X: uint32,Y: uint32,R: set_Pr1773385645901665561uint32,Xs: list_uint32] :
      ( ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y ) @ R )
     => ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ ( cons_uint32 @ X @ Xs ) @ ( cons_uint32 @ Y @ Xs ) ) @ ( listrel1_uint32 @ R ) ) ) ).

% listrel1I1
thf(fact_7004_listrel1I1,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_7005_listrel1I1,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Xs ) ) @ ( listrel1_int @ R ) ) ) ).

% listrel1I1
thf(fact_7006_Cons__listrel1E1,axiom,
    ! [X: $o,Xs: list_o,Ys: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs ) @ Ys ) @ ( listrel1_o @ R ) )
     => ( ! [Y3: $o] :
            ( ( Ys
              = ( cons_o @ Y3 @ Xs ) )
           => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_o] :
              ( ( Ys
                = ( cons_o @ X @ Zs2 ) )
             => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Zs2 ) @ ( listrel1_o @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_7007_Cons__listrel1E1,axiom,
    ! [X: uint32,Xs: list_uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ ( cons_uint32 @ X @ Xs ) @ Ys ) @ ( listrel1_uint32 @ R ) )
     => ( ! [Y3: uint32] :
            ( ( Ys
              = ( cons_uint32 @ Y3 @ Xs ) )
           => ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_uint32] :
              ( ( Ys
                = ( cons_uint32 @ X @ Zs2 ) )
             => ~ ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Xs @ Zs2 ) @ ( listrel1_uint32 @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_7008_Cons__listrel1E1,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( Ys
              = ( cons_nat @ Y3 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys
                = ( cons_nat @ X @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_7009_Cons__listrel1E1,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys ) @ ( listrel1_int @ R ) )
     => ( ! [Y3: int] :
            ( ( Ys
              = ( cons_int @ Y3 @ Xs ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_int] :
              ( ( Ys
                = ( cons_int @ X @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_7010_Cons__listrel1E2,axiom,
    ! [Xs: list_o,Y: $o,Ys: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ ( cons_o @ Y @ Ys ) ) @ ( listrel1_o @ R ) )
     => ( ! [X3: $o] :
            ( ( Xs
              = ( cons_o @ X3 @ Ys ) )
           => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_o] :
              ( ( Xs
                = ( cons_o @ Y @ Zs2 ) )
             => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Zs2 @ Ys ) @ ( listrel1_o @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_7011_Cons__listrel1E2,axiom,
    ! [Xs: list_uint32,Y: uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Xs @ ( cons_uint32 @ Y @ Ys ) ) @ ( listrel1_uint32 @ R ) )
     => ( ! [X3: uint32] :
            ( ( Xs
              = ( cons_uint32 @ X3 @ Ys ) )
           => ~ ( member8027108493173000802uint32 @ ( produc1400373151660368625uint32 @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_uint32] :
              ( ( Xs
                = ( cons_uint32 @ Y @ Zs2 ) )
             => ~ ( member2333554998283850498uint32 @ ( produc7487160679990061969uint32 @ Zs2 @ Ys ) @ ( listrel1_uint32 @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_7012_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
     => ( ! [X3: nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_7013_Cons__listrel1E2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R ) )
     => ( ! [X3: int] :
            ( ( Xs
              = ( cons_int @ X3 @ Ys ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_int] :
              ( ( Xs
                = ( cons_int @ Y @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_7014_sum__bounded__below,axiom,
    ! [A4: set_VEBT_VEBT,K4: real,F: vEBT_VEBT > real] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A4 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite7802652506058667612T_VEBT @ A4 ) ) @ K4 ) @ ( groups2240296850493347238T_real @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7015_sum__bounded__below,axiom,
    ! [A4: set_real,K4: real,F: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A4 ) ) @ K4 ) @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7016_sum__bounded__below,axiom,
    ! [A4: set_int,K4: real,F: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A4 ) ) @ K4 ) @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7017_sum__bounded__below,axiom,
    ! [A4: set_complex,K4: real,F: complex > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A4 ) ) @ K4 ) @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7018_sum__bounded__below,axiom,
    ! [A4: set_literal,K4: real,F: literal > real] :
      ( ! [I2: literal] :
          ( ( member_literal @ I2 @ A4 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_literal @ A4 ) ) @ K4 ) @ ( groups2795228986978005958l_real @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7019_sum__bounded__below,axiom,
    ! [A4: set_VEBT_VEBT,K4: code_integer,F: vEBT_VEBT > code_integer] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I2 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite7802652506058667612T_VEBT @ A4 ) ) @ K4 ) @ ( groups5748017345553531991nteger @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7020_sum__bounded__below,axiom,
    ! [A4: set_real,K4: code_integer,F: real > code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I2 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_real @ A4 ) ) @ K4 ) @ ( groups7713935264441627589nteger @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7021_sum__bounded__below,axiom,
    ! [A4: set_int,K4: code_integer,F: int > code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I2 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A4 ) ) @ K4 ) @ ( groups7873554091576472773nteger @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7022_sum__bounded__below,axiom,
    ! [A4: set_complex,K4: code_integer,F: complex > code_integer] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I2 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_complex @ A4 ) ) @ K4 ) @ ( groups6621422865394947399nteger @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7023_sum__bounded__below,axiom,
    ! [A4: set_literal,K4: code_integer,F: literal > code_integer] :
      ( ! [I2: literal] :
          ( ( member_literal @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I2 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_literal @ A4 ) ) @ K4 ) @ ( groups8556826181031883383nteger @ F @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_7024_sum__bounded__above,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real,K4: real] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A4 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite7802652506058667612T_VEBT @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7025_sum__bounded__above,axiom,
    ! [A4: set_real,F: real > real,K4: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7026_sum__bounded__above,axiom,
    ! [A4: set_int,F: int > real,K4: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7027_sum__bounded__above,axiom,
    ! [A4: set_complex,F: complex > real,K4: real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7028_sum__bounded__above,axiom,
    ! [A4: set_literal,F: literal > real,K4: real] :
      ( ! [I2: literal] :
          ( ( member_literal @ I2 @ A4 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups2795228986978005958l_real @ F @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_literal @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7029_sum__bounded__above,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > code_integer,K4: code_integer] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K4 ) )
     => ( ord_le3102999989581377725nteger @ ( groups5748017345553531991nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite7802652506058667612T_VEBT @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7030_sum__bounded__above,axiom,
    ! [A4: set_real,F: real > code_integer,K4: code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K4 ) )
     => ( ord_le3102999989581377725nteger @ ( groups7713935264441627589nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_real @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7031_sum__bounded__above,axiom,
    ! [A4: set_int,F: int > code_integer,K4: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K4 ) )
     => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7032_sum__bounded__above,axiom,
    ! [A4: set_complex,F: complex > code_integer,K4: code_integer] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K4 ) )
     => ( ord_le3102999989581377725nteger @ ( groups6621422865394947399nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_complex @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7033_sum__bounded__above,axiom,
    ! [A4: set_literal,F: literal > code_integer,K4: code_integer] :
      ( ! [I2: literal] :
          ( ( member_literal @ I2 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K4 ) )
     => ( ord_le3102999989581377725nteger @ ( groups8556826181031883383nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_literal @ A4 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_7034_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N6: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N6 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).

% nat_less_real_le
thf(fact_7035_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N6: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N6 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_7036_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_7037_lenlex__length,axiom,
    ! [Ms: list_VEBT_VEBT,Ns: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Ms @ Ns ) @ ( lenlex_VEBT_VEBT @ R ) )
     => ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Ms ) @ ( size_s6755466524823107622T_VEBT @ Ns ) ) ) ).

% lenlex_length
thf(fact_7038_lenlex__length,axiom,
    ! [Ms: list_real,Ns: list_real,R: set_Pr6218003697084177305l_real] :
      ( ( member6584958104391596930t_real @ ( produc1408950526243324945t_real @ Ms @ Ns ) @ ( lenlex_real @ R ) )
     => ( ord_less_eq_nat @ ( size_size_list_real @ Ms ) @ ( size_size_list_real @ Ns ) ) ) ).

% lenlex_length
thf(fact_7039_lenlex__length,axiom,
    ! [Ms: list_o,Ns: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ms @ Ns ) @ ( lenlex_o @ R ) )
     => ( ord_less_eq_nat @ ( size_size_list_o @ Ms ) @ ( size_size_list_o @ Ns ) ) ) ).

% lenlex_length
thf(fact_7040_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_7041_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ! [M3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M3 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C2 ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_7042_sum__bounded__above__strict,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > int,K4: int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A4 )
         => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite7802652506058667612T_VEBT @ A4 ) )
       => ( ord_less_int @ ( groups769130701875090982BT_int @ F @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite7802652506058667612T_VEBT @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7043_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F: real > int,K4: int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A4 ) )
       => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7044_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F: complex > int,K4: int] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
       => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7045_sum__bounded__above__strict,axiom,
    ! [A4: set_literal,F: literal > int,K4: int] :
      ( ! [I2: literal] :
          ( ( member_literal @ I2 @ A4 )
         => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ A4 ) )
       => ( ord_less_int @ ( groups8649609317433967686al_int @ F @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_literal @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7046_sum__bounded__above__strict,axiom,
    ! [A4: set_nat,F: nat > int,K4: int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A4 )
         => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7047_sum__bounded__above__strict,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > code_integer,K4: code_integer] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A4 )
         => ( ord_le6747313008572928689nteger @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite7802652506058667612T_VEBT @ A4 ) )
       => ( ord_le6747313008572928689nteger @ ( groups5748017345553531991nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite7802652506058667612T_VEBT @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7048_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F: real > code_integer,K4: code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ord_le6747313008572928689nteger @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A4 ) )
       => ( ord_le6747313008572928689nteger @ ( groups7713935264441627589nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_real @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7049_sum__bounded__above__strict,axiom,
    ! [A4: set_int,F: int > code_integer,K4: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ord_le6747313008572928689nteger @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
       => ( ord_le6747313008572928689nteger @ ( groups7873554091576472773nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7050_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F: complex > code_integer,K4: code_integer] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ord_le6747313008572928689nteger @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
       => ( ord_le6747313008572928689nteger @ ( groups6621422865394947399nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_complex @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7051_sum__bounded__above__strict,axiom,
    ! [A4: set_literal,F: literal > code_integer,K4: code_integer] :
      ( ! [I2: literal] :
          ( ( member_literal @ I2 @ A4 )
         => ( ord_le6747313008572928689nteger @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ A4 ) )
       => ( ord_le6747313008572928689nteger @ ( groups8556826181031883383nteger @ F @ A4 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_literal @ A4 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_7052_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_7053_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( vEBT_VEBT_cnt2 @ T2 ) ) ) ) ).

% cnt_cnt_eq
thf(fact_7054_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_VEBT_minNull @ T ) ) ) ) ) ).

% minNulli_rule
thf(fact_7055_builupi_Hcorr,axiom,
    ! [N: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) ) ).

% builupi'corr
thf(fact_7056_builupicorr,axiom,
    ! [N: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) ) ).

% builupicorr
thf(fact_7057_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts2 ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_7058_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts2 ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_minti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_7059_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_7060_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_7061_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) ) ) ).

% vebt_minti_h
thf(fact_7062_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) ) ) ).

% vebt_maxti_h
thf(fact_7063_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z2: int] :
        ? [N6: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N6 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_7064_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_7065_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_7066_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7067_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N2: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_7068_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_7069_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_7070_int__of__nat__induct,axiom,
    ! [P2: int > $o,Z: int] :
      ( ! [N2: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P2 @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_7071_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_7072_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_7073_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_7074_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_7075_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_7076_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_7077_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_less_as_int
thf(fact_7078_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_leq_as_int
thf(fact_7079_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_7080_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_7081_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_7082_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_7083_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_7084_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_7085_zdiff__int__split,axiom,
    ! [P2: int > $o,X: nat,Y: nat] :
      ( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P2 @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_7086_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_7087_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_7088_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_7089_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_VEBT_minNull @ T ) ) ) )
      @ one_one_nat ) ).

% minNrulli_ruleT
thf(fact_7090_T__vebt__buildupi,axiom,
    ! [N: nat,H2: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% T_vebt_buildupi
thf(fact_7091_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_hT
thf(fact_7092_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_hT
thf(fact_7093_htt__vebt__buildupi_H,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% htt_vebt_buildupi'
thf(fact_7094_htt__vebt__buildupi,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% htt_vebt_buildupi
thf(fact_7095_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_7096_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_7097_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_7098_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_7099_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_7100_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% zle_add1_eq_le
thf(fact_7101_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% zle_diff1_eq
thf(fact_7102_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_7103_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_7104_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_7105_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_7106_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z2: int] :
        ? [N6: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N6 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_7107_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_7108_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_7109_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_7110_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_7111_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z )
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
thf(fact_7112_int__gr__induct,axiom,
    ! [K: int,I: int,P2: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K @ I2 )
             => ( ( P2 @ I2 )
               => ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_gr_induct
thf(fact_7113_int__less__induct,axiom,
    ! [I: int,K: int,P2: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K )
             => ( ( P2 @ I2 )
               => ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_less_induct
thf(fact_7114_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_7115_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P2 @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_12: int] : ( P6 @ X_12 )
           => ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_7116_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P2 @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_7117_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_7118_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_7119_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
      = ( ord_less_int @ W2 @ Z ) ) ).

% add1_zle_eq
thf(fact_7120_int__ge__induct,axiom,
    ! [K: int,I: int,P2: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P2 @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P2 @ I2 )
               => ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_ge_induct
thf(fact_7121_int__le__induct,axiom,
    ! [I: int,K: int,P2: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P2 @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P2 @ I2 )
               => ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_le_induct
thf(fact_7122_int__induct,axiom,
    ! [P2: int > $o,K: int,I: int] :
      ( ( P2 @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P2 @ I2 )
             => ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P2 @ I2 )
               => ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_induct
thf(fact_7123_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q7: int,R6: int,Q3: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q7 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( ord_less_int @ B @ R6 )
           => ( ord_less_eq_int @ Q3 @ Q7 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_7124_unique__quotient__lemma,axiom,
    ! [B: int,Q7: int,R6: int,Q3: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q7 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
       => ( ( ord_less_int @ R6 @ B )
         => ( ( ord_less_int @ R @ B )
           => ( ord_less_eq_int @ Q7 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_7125_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R: int,B2: int,Q7: int,R6: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R6 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R6 ) @ zero_zero_int )
       => ( ( ord_less_int @ R @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q7 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_7126_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R: int,B2: int,Q7: int,R6: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R6 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R6 ) )
       => ( ( ord_less_int @ R6 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q3 @ Q7 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_7127_q__pos__lemma,axiom,
    ! [B2: int,Q7: int,R6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R6 ) )
     => ( ( ord_less_int @ R6 @ B2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ord_less_eq_int @ zero_zero_int @ Q7 ) ) ) ) ).

% q_pos_lemma
thf(fact_7128_incr__mult__lemma,axiom,
    ! [D: int,P2: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P2 @ X3 )
           => ( P2 @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P2 @ X5 )
             => ( P2 @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_7129_pos__mult__pos__ge,axiom,
    ! [X: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ ( times_times_int @ N @ one_one_int ) @ ( times_times_int @ N @ X ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_7130_decr__mult__lemma,axiom,
    ! [D: int,P2: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P2 @ X5 )
             => ( P2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_7131_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_7132_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( ( ord_less_eq_int @ N @ M )
     => ( ord_less_int @ ( minus_minus_int @ N @ one_one_int ) @ M ) ) ).

% less_1_helper
thf(fact_7133_imp__le__cong,axiom,
    ! [X: int,X9: int,P2: $o,P6: $o] :
      ( ( X = X9 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
         => ( P2 = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P2 )
          = ( ( ord_less_eq_int @ zero_zero_int @ X9 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_7134_conj__le__cong,axiom,
    ! [X: int,X9: int,P2: $o,P6: $o] :
      ( ( X = X9 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
         => ( P2 = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P2 )
          = ( ( ord_less_eq_int @ zero_zero_int @ X9 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_7135_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_7136_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_7137_Gcd__int__greater__eq__0,axiom,
    ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_7138_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_7139_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N ) )
      = ( vEBT_V9176841429113362141ildupi @ N ) ) ).

% Tbuildupi_buildupi'
thf(fact_7140_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_7141_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_7142_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = C2 ) ) ) ) ).

% finite_nth_roots
thf(fact_7143_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X3: real,Y3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
           => ( X3 = Y ) )
       => ( ( foldr_real_real @ plus_plus_real @ Xs @ zero_zero_real )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs ) ) @ Y ) ) ) ) ).

% foldr_same
thf(fact_7144_vebt__maxti__rule,axiom,
    ! [N: nat,S: set_nat,Ti: vEBT_VEBTi,Y: nat] :
      ( time_htt_option_nat @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
          @ ( pure_assn
            @ ( ( R5
                = ( some_nat @ Y ) )
              = ( vEBT_VEBT_max_in_set @ S @ Y ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_rule
thf(fact_7145_foldr0,axiom,
    ! [Xs: list_real,C2: real,D: real] :
      ( ( foldr_real_real @ plus_plus_real @ Xs @ ( plus_plus_real @ C2 @ D ) )
      = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs @ D ) @ C2 ) ) ).

% foldr0
thf(fact_7146_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_7147_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( 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_7148_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z2: complex] :
                  ( ( power_power_complex @ Z2 @ N )
                  = C2 ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_7149_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_7150_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_7151_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C2 ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_7152_vebt__buildupi__rule__basic,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ bot_bot_set_nat ) ) ) ).

% vebt_buildupi_rule_basic
thf(fact_7153_vebt__minti__rule,axiom,
    ! [N: nat,S: set_nat,Ti: vEBT_VEBTi,Y: nat] :
      ( time_htt_option_nat @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
          @ ( pure_assn
            @ ( ( R5
                = ( some_nat @ Y ) )
              = ( vEBT_VEBT_min_in_set @ S @ Y ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_rule
thf(fact_7154_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( 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_7155_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_7156_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( 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_7157_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_real )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( 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_7158_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_7159_vebt__buildupi__refines,axiom,
    ! [N: nat] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_V739175172307565963ildupi @ N ) ) ).

% vebt_buildupi_refines
thf(fact_7160_vebt__deletei__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_V1365221501068881998eletei @ T @ Ti @ X ) ) ).

% vebt_deletei_refines
thf(fact_7161_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) ) ).

% vebt_inserti_refines
thf(fact_7162_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_7163_vebt__succi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_succi @ Ti @ X ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X ) ) ).

% vebt_succi_refines
thf(fact_7164_vebt__predi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_predi @ Ti @ X ) @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X ) ) ).

% vebt_predi_refines
thf(fact_7165_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine_Imp_refines_o @ ( vEBT_vebt_memberi @ Ti @ X ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X ) ) ).

% vebt_memberi_refines
thf(fact_7166_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_7167_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ one_one_nat ) ) ).

% TBOUND_minNull
thf(fact_7168_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% TBOUND_vebt_buildupi
thf(fact_7169_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_7170_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_7171_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_7172_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_7173_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_7174_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_7175_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_7176_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_7177_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_minti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_minti
thf(fact_7178_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_maxti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_maxti
thf(fact_7179_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : ( time_TBOUND_o @ ( vEBT_VEBT_minNulli @ T ) @ one_one_nat ) ).

% TBOUND_minNulli
thf(fact_7180_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_7181_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_7182_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).

% card_atLeastAtMost
thf(fact_7183_all__nat__less,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
           => ( P2 @ M5 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less
thf(fact_7184_ex__nat__less,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
            & ( P2 @ M5 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less
thf(fact_7185_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_7186_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_7187_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_7188_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_7189_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_7190_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( set_or1269000886237332187st_nat @ M @ N )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_7191_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_7192_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_7193_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N8: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N8 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N8 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_7194_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_7195_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_7196_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_7197_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_7198_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_7199_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_7200_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X5
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X5 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_7201_aset_I5_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X5
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X5 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_7202_aset_I4_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X5
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( plus_plus_int @ X5 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_7203_aset_I3_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X5
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( plus_plus_int @ X5 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_7204_bset_I7_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X5 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_7205_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X5 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_7206_bset_I4_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( minus_minus_int @ X5 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_7207_bset_I3_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( minus_minus_int @ X5 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_7208_periodic__finite__ex,axiom,
    ! [D: int,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P2 @ X3 )
            = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X8: int] : ( P2 @ X8 ) )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_7209_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X5
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X5 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_7210_aset_I6_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X5
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X5 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_7211_bset_I8_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X5 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_7212_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X5 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_7213_cpmi,axiom,
    ! [D4: int,P2: int > $o,P6: int > $o,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P2 @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ B4 )
                     => ( X3
                       != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P2 @ X8 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ B4 )
                        & ( P2 @ ( plus_plus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_7214_cppi,axiom,
    ! [D4: int,P2: int > $o,P6: int > $o,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P2 @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ A4 )
                     => ( X3
                       != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P2 @ X8 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ A4 )
                        & ( P2 @ ( minus_minus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_7215_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_7216_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_7217_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_7218_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_7219_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_7220_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_7221_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_7222_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_7223_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_7224_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_7225_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_7226_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_7227_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_7228_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_7229_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_7230_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_7231_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_7232_zdiv__mono2__neg,axiom,
    ! [A: int,B2: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_7233_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_7234_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_7235_zdiv__mono2,axiom,
    ! [A: int,B2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_7236_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_7237_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_7238_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_7239_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
        = ( plus_plus_nat @ ( divide_divide_nat @ M @ N ) @ K ) ) ) ).

% msrevs(1)
thf(fact_7240_verit__less__mono__div__int2,axiom,
    ! [A4: int,B4: int,N: int] :
      ( ( ord_less_eq_int @ A4 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A4 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_7241_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_7242_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_pos_eq
thf(fact_7243_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_neg_eq
thf(fact_7244_split__zdiv,axiom,
    ! [P2: int > $o,N: int,K: int] :
      ( ( P2 @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P2 @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P2 @ I3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P2 @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_7245_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_7246_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_7247_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_7248_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_7249_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_7250_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_7251_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N )
          = M )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_7252_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_7253_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_7254_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_7255_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_7256_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_7257_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_7258_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_7259_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_7260_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_7261_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_7262_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_7263_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_7264_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ N @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_7265_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_7266_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).

% div_less_dividend
thf(fact_7267_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_7268_td__gal__lt,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_nat @ A @ ( times_times_nat @ B @ C2 ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A @ C2 ) @ B ) ) ) ).

% td_gal_lt
thf(fact_7269_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_7270_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).

% zdiv_zmult2_eq
thf(fact_7271_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N6: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M5 @ N6 )
            | ( N6 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N6 ) @ N6 ) ) ) ) ) ).

% div_if
thf(fact_7272_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_7273_split__div,axiom,
    ! [P2: nat > $o,M: nat,N: nat] :
      ( ( P2 @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P2 @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
               => ( P2 @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_7274_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_7275_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_7276_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_7277_td__gal,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B @ C2 ) @ A )
        = ( ord_less_eq_nat @ B @ ( divide_divide_nat @ A @ C2 ) ) ) ) ).

% td_gal
thf(fact_7278_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N6: nat] :
          ( if_nat @ ( N6 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N6 ) @ M5 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_7279_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_7280_split__div_H,axiom,
    ! [P2: nat > $o,M: nat,N: nat] :
      ( ( P2 @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P2 @ zero_zero_nat ) )
        | ? [Q8: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q8 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q8 ) ) )
            & ( P2 @ Q8 ) ) ) ) ).

% split_div'
thf(fact_7281_power__sub,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_sub
thf(fact_7282_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_7283_nat__mod__eq_H,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ A @ N )
     => ( ( modulo_modulo_nat @ A @ N )
        = A ) ) ).

% nat_mod_eq'
thf(fact_7284_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_7285_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_7286_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_7287_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_7288_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_7289_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_7290_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_7291_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_7292_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_7293_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_7294_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_7295_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_7296_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_7297_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% mod_Suc_eq
thf(fact_7298_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_7299_nat__mod__eq,axiom,
    ! [B: nat,N: nat,A: nat] :
      ( ( ord_less_nat @ B @ N )
     => ( ( ( modulo_modulo_nat @ A @ N )
          = ( modulo_modulo_nat @ B @ N ) )
       => ( ( modulo_modulo_nat @ A @ N )
          = B ) ) ) ).

% nat_mod_eq
thf(fact_7300_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_7301_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_7302_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).

% Suc_times_binomial
thf(fact_7303_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_7304_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_7305_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).

% binomial_le_pow
thf(fact_7306_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_7307_mod__induct,axiom,
    ! [P2: nat > $o,N: nat,P3: nat,M: nat] :
      ( ( P2 @ N )
     => ( ( ord_less_nat @ N @ P3 )
       => ( ( ord_less_nat @ M @ P3 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P3 )
               => ( ( P2 @ N2 )
                 => ( P2 @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P3 ) ) ) )
           => ( P2 @ M ) ) ) ) ) ).

% mod_induct
thf(fact_7308_gcd__nat__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N: nat] :
      ( ! [M3: nat] : ( P2 @ M3 @ zero_zero_nat )
     => ( ! [M3: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P2 @ N2 @ ( modulo_modulo_nat @ M3 @ N2 ) )
             => ( P2 @ M3 @ N2 ) ) )
       => ( P2 @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_7309_nat__mod__lem,axiom,
    ! [N: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ B @ N )
        = ( ( modulo_modulo_nat @ B @ N )
          = B ) ) ) ).

% nat_mod_lem
thf(fact_7310_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_7311_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_7312_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N: nat] :
      ( ( ( plus_plus_nat @ L @ K )
        = ( plus_plus_nat @ D @ ( modulo_modulo_nat @ K @ L ) ) )
     => ( ( ord_less_nat @ N @ L )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ D @ N ) @ L )
          = N ) ) ) ).

% word_rot_lem
thf(fact_7313_nat__minus__mod,axiom,
    ! [N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ N @ ( modulo_modulo_nat @ N @ M ) ) @ M )
      = zero_zero_nat ) ).

% nat_minus_mod
thf(fact_7314_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_7315_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N6: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N6 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N6 ) @ N6 ) ) ) ) ).

% mod_if
thf(fact_7316_mod__nat__sub,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( modulo_modulo_nat @ ( minus_minus_nat @ X @ Y ) @ Z )
        = ( minus_minus_nat @ X @ Y ) ) ) ).

% mod_nat_sub
thf(fact_7317_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q5: nat] :
          ( M
          = ( times_times_nat @ D @ Q5 ) ) ) ).

% mod_eq_0D
thf(fact_7318_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_7319_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_7320_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_7321_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_7322_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_7323_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_7324_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_7325_div__less__mono,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A4 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B4 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_7326_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ( ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( plus_plus_nat @ X @ Y ) ) )
          & ( ~ ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( minus_minus_nat @ ( plus_plus_nat @ X @ Y ) @ Z ) ) ) ) ) ) ).

% mod_nat_add
thf(fact_7327_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q5: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q5 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_7328_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S4: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S4 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_7329_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S4: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S4 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_7330_sum__choose__upper,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_7331_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_7332_split__mod,axiom,
    ! [P2: nat > $o,M: nat,N: nat] :
      ( ( P2 @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P2 @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
               => ( P2 @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_7333_mod__lemma,axiom,
    ! [C2: nat,R: nat,B: nat,Q3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_nat @ R @ B )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ Q3 @ C2 ) ) @ R ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mod_lemma
thf(fact_7334_sum__choose__lower,axiom,
    ! [R: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_7335_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_7336_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_7337_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_7338_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_7339_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_7340_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B4 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_7341_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_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) ) ).

% vebt_assn_raw.simps(2)
thf(fact_7342_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_7343_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_7344_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_7345_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_7346_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_7347_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_7348_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_7349_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_7350_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q8: int] :
            ( M
            = ( times_times_int @ D @ Q8 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_7351_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q5: int] :
          ( M
          = ( times_times_int @ D @ Q5 ) ) ) ).

% zmod_eq_0D
thf(fact_7352_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_7353_int__mod__eq,axiom,
    ! [B: int,N: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ N )
       => ( ( ( modulo_modulo_int @ A @ N )
            = ( modulo_modulo_int @ B @ N ) )
         => ( ( modulo_modulo_int @ A @ N )
            = B ) ) ) ) ).

% int_mod_eq
thf(fact_7354_int__mod__lem,axiom,
    ! [N: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ N )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ B )
          & ( ord_less_int @ B @ N ) )
        = ( ( modulo_modulo_int @ B @ N )
          = B ) ) ) ).

% int_mod_lem
thf(fact_7355_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_7356_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_7357_int__mod__ge,axiom,
    ! [A: int,N: int] :
      ( ( ord_less_int @ A @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ A @ ( modulo_modulo_int @ A @ N ) ) ) ) ).

% int_mod_ge
thf(fact_7358_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_7359_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_7360_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_7361_int__mod__le_H,axiom,
    ! [B: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ B @ N ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ B @ N ) @ ( minus_minus_int @ B @ N ) ) ) ).

% int_mod_le'
thf(fact_7362_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_7363_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_7364_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_7365_zdiv__mono__strict,axiom,
    ! [A4: int,B4: int,N: int] :
      ( ( ord_less_int @ A4 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A4 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B4 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_7366_int__mod__ge_H,axiom,
    ! [B: int,N: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ ( plus_plus_int @ B @ N ) @ ( modulo_modulo_int @ B @ N ) ) ) ) ).

% int_mod_ge'
thf(fact_7367_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_7368_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_7369_mod__power__lem,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( ord_less_eq_nat @ M @ N )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ M ) )
            = zero_zero_int ) )
        & ( ~ ( ord_less_eq_nat @ M @ N )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ M ) )
            = ( power_power_int @ A @ N ) ) ) ) ) ).

% mod_power_lem
thf(fact_7370_split__zmod,axiom,
    ! [P2: int > $o,N: int,K: int] :
      ( ( P2 @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P2 @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P2 @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P2 @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_7371_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_neg_eq
thf(fact_7372_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_pos_eq
thf(fact_7373_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_7374_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_7375_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ ( plus_plus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
thf(fact_7376_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ ( minus_minus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
thf(fact_7377_zmod__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_7378_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_7379_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_7380_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,N: int] :
      ( ( ord_less_int @ A4 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B4 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B4 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_7381_split__neg__lemma,axiom,
    ! [K: int,P2: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P2 @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P2 @ I3 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_7382_split__pos__lemma,axiom,
    ! [K: int,P2: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P2 @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P2 @ I3 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_7383_vebt__assn__raw_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xa2
                  = ( vEBT_Leafi @ Ai @ Bi ) )
               => ( Y
                 != ( pure_assn
                    @ ( ( Ai = A3 )
                      & ( Bi = B3 ) ) ) ) ) )
       => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
             => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                  ( ( Xa2
                    = ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
                 => ( Y
                   != ( times_times_assn
                      @ ( times_times_assn
                        @ ( pure_assn
                          @ ( ( Mmoi2 = Mmo2 )
                            & ( Degi2 = Deg2 ) ) )
                        @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
                      @ ( ex_ass463751140784270563_VEBTi
                        @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) ) ) )
         => ( ( ? [V: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) )
             => ( ? [Vd3: $o,Ve3: $o] :
                    ( Xa2
                    = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] :
                    ( X
                    = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
               => ( ? [V: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                      ( Xa2
                      = ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
thf(fact_7384_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_7385_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_7386_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_7387_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_less_mono
thf(fact_7388_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel_iff
thf(fact_7389_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_7390_one__less__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_less_exp_iff
thf(fact_7391_exp__less__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_7392_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_7393_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_7394_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel
thf(fact_7395_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_7396_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] :
          ( Y
         != ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X222: $o] :
            ( Y
           != ( vEBT_Leafi @ X212 @ X222 ) ) ) ).

% VEBTi.exhaust
thf(fact_7397_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_7398_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_7399_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y ) ) ).

% exp_total
thf(fact_7400_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_7401_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_7402_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_7403_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_7404_exp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).

% exp_gt_one
thf(fact_7405_exp__ge__add__one__self,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).

% exp_ge_add_one_self
thf(fact_7406_vebt__assn__raw_Ocases,axiom,
    ! [X: produc3625547720036274456_VEBTi] :
      ( ! [A3: $o,B3: $o,Ai: $o,Bi: $o] :
          ( X
         != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B3 ) @ ( 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] :
            ( X
           != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) )
       => ( ! [V: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
              ( X
             != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
         => ~ ! [Vd3: $o,Ve3: $o,V: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                ( X
               != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ).

% vebt_assn_raw.cases
thf(fact_7407_VEBT__internal_OminNulli_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ( X
       != ( vEBT_Leafi @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leafi @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leafi @ Uu @ $true ) )
         => ( ! [Uw: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] :
                ( X
               != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                  ( X
                 != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNulli.cases
thf(fact_7408_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_7409_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V2: option4927543243414619207at_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd2: $o,Ve2: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V2 @ Va2 @ Vb @ Vc ) @ ( vEBT_Leafi @ Vd2 @ Ve2 ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(3)
thf(fact_7410_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_7411_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_7412_vebt__minti_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leafi @ A3 @ B3 ) )
     => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
            ( X
           != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
              ( X
             != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ).

% vebt_minti.cases
thf(fact_7413_vebt__assn__def,axiom,
    ( vEBT_Intf_vebt_assn
    = ( ^ [N6: nat,S8: set_nat,Ti2: vEBT_VEBTi] :
          ( ex_assn_VEBT_VEBT
          @ ^ [T2: vEBT_VEBT] :
              ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti2 )
              @ ( pure_assn
                @ ( ( S8
                    = ( vEBT_set_vebt @ T2 ) )
                  & ( vEBT_invar_vebt @ T2 @ N6 ) ) ) ) ) ) ) ).

% vebt_assn_def
thf(fact_7414_vebt__assn__raw_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa2 )
        = Y )
     => ( ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xa2
                    = ( vEBT_Leafi @ Ai @ Bi ) )
                 => ( ( Y
                      = ( pure_assn
                        @ ( ( Ai = A3 )
                          & ( Bi = B3 ) ) ) )
                   => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
         => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
               => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                    ( ( Xa2
                      = ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
                   => ( ( Y
                        = ( times_times_assn
                          @ ( times_times_assn
                            @ ( pure_assn
                              @ ( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ) )
                            @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
                          @ ( ex_ass463751140784270563_VEBTi
                            @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) )
                     => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) ) ) ) )
           => ( ! [V: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xa2
                        = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
                     => ( ( Y = bot_bot_assn )
                       => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( X
                      = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
                   => ! [V: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa2
                          = ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) )
                       => ( ( Y = bot_bot_assn )
                         => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
thf(fact_7415_vebt__minti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leafi @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                    & ( ~ B3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) ) ) ) ) ) ).

% vebt_minti.elims
thf(fact_7416_vebt__maxti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leafi @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                    & ( ~ A3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) ) ) ) ) ) ).

% vebt_maxti.elims
thf(fact_7417_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_7418_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_7419_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_7420_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_Suc_0_nat
thf(fact_7421_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).

% fact_div_fact_le_pow
thf(fact_7422_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_7423_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_7424_prod__Suc__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_Suc_fact
thf(fact_7425_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_minti.simps(2)
thf(fact_7426_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_maxti.simps(2)
thf(fact_7427_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_7428_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi ) ) ) ).

% vebt_minti.simps(3)
thf(fact_7429_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma ) ) ) ).

% vebt_maxti.simps(3)
thf(fact_7430_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_7431_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_7432_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_7433_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_7434_vebt__maxti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leafi @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                      & ( ~ A3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxti.pelims
thf(fact_7435_vebt__minti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leafi @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                      & ( ~ B3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_minti.pelims
thf(fact_7436_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_7437_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_7438_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_7439_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
            & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_7440_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_7441_Least__eq__0,axiom,
    ! [P2: nat > $o] :
      ( ( P2 @ zero_zero_nat )
     => ( ( ord_Least_nat @ P2 )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_7442_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_7443_artanh__minus__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
        = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).

% artanh_minus_real
thf(fact_7444_Least__Suc2,axiom,
    ! [P2: nat > $o,N: nat,Q2: nat > $o,M: nat] :
      ( ( P2 @ N )
     => ( ( Q2 @ M )
       => ( ~ ( P2 @ zero_zero_nat )
         => ( ! [K2: nat] :
                ( ( P2 @ ( suc @ K2 ) )
                = ( Q2 @ K2 ) )
           => ( ( ord_Least_nat @ P2 )
              = ( suc @ ( ord_Least_nat @ Q2 ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_7445_Least__Suc,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ( ( ord_Least_nat @ P2 )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M5: nat] : ( P2 @ ( suc @ M5 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_7446_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A7: real] : ( if_real @ ( ord_less_real @ A7 @ zero_zero_real ) @ ( uminus_uminus_real @ A7 ) @ A7 ) ) ) ).

% abs_real_def
thf(fact_7447_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V2: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V2 )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V2 ) ) ) ).

% sin_bound_lemma
thf(fact_7448_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D2 )
               => ( ( ord_less_eq_real @ A @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_7449_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D2 )
               => ( ( ord_less_real @ A @ Y5 )
                  & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_7450_set__vebt__succ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ? [X5: nat] :
              ( ( member_nat @ X5 @ ( vEBT_set_vebt @ T ) )
              & ( ord_less_nat @ X @ X5 ) )
         => ( ( vEBT_vebt_succ @ T @ X )
            = ( some_nat
              @ ( ord_Least_nat
                @ ^ [Y4: nat] :
                    ( ( member_nat @ Y4 @ ( vEBT_set_vebt @ T ) )
                    & ( ord_less_nat @ X @ Y4 ) ) ) ) ) )
        & ( ~ ? [X3: nat] :
                ( ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) )
                & ( ord_less_nat @ X @ X3 ) )
         => ( ( vEBT_vebt_succ @ T @ X )
            = none_nat ) ) ) ) ).

% set_vebt_succ'
thf(fact_7451_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_7452_cos__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( cos_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% cos_coeff_Suc
thf(fact_7453_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_7454_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_arctan_iff
thf(fact_7455_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_7456_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_7457_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_7458_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_7459_set__vebt__pred_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ? [X5: nat] :
              ( ( member_nat @ X5 @ ( vEBT_set_vebt @ T ) )
              & ( ord_less_nat @ X5 @ X ) )
         => ( ( vEBT_vebt_pred @ T @ X )
            = ( some_nat
              @ ( order_Greatest_nat
                @ ^ [Y4: nat] :
                    ( ( member_nat @ Y4 @ ( vEBT_set_vebt @ T ) )
                    & ( ord_less_nat @ Y4 @ X ) ) ) ) ) )
        & ( ~ ? [X3: nat] :
                ( ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) )
                & ( ord_less_nat @ X3 @ X ) )
         => ( ( vEBT_vebt_pred @ T @ X )
            = none_nat ) ) ) ) ).

% set_vebt_pred'
thf(fact_7460_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% arctan_less_iff
thf(fact_7461_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone
thf(fact_7462_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_7463_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% arctan_le_iff
thf(fact_7464_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_7465_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_7466_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ! [M5: int] :
          ? [N6: int] :
            ( ( ord_less_eq_int @ M5 @ ( abs_abs_int @ N6 ) )
            & ( member_int @ N6 @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_7467_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ! [M5: int] :
          ? [N6: int] :
            ( ( ord_less_int @ M5 @ ( abs_abs_int @ N6 ) )
            & ( member_int @ N6 @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_7468_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).

% zabs_def
thf(fact_7469_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_7470_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M @ I2 )
            & ( ord_less_nat @ I2 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M @ I2 )
                & ( ord_less_eq_nat @ I2 @ N )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_7471_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_7472_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_7473_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_7474_sin__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( sin_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% sin_coeff_Suc
thf(fact_7475_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7476_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_7477_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_7478_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ( ln_ln_real @ X )
            = ( ln_ln_real @ Y ) )
          = ( X = Y ) ) ) ) ).

% ln_inj_iff
thf(fact_7479_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_7480_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_7481_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = zero_zero_real )
        = ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_7482_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_iff
thf(fact_7483_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_7484_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_7485_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X ) )
        = X )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% exp_ln_iff
thf(fact_7486_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_iff
thf(fact_7487_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_7488_ln__less__self,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_less_self
thf(fact_7489_GreatestI__ex__nat,axiom,
    ! [P2: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P2 @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_7490_Greatest__le__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% Greatest_le_nat
thf(fact_7491_GreatestI__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_nat
thf(fact_7492_ln__bound,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_bound
thf(fact_7493_ln__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_gt_zero
thf(fact_7494_ln__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_7495_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_7496_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_7497_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_7498_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_7499_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_7500_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = ( minus_minus_real @ X @ one_one_real ) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_7501_ln__div,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_7502_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).

% ln_ge_iff
thf(fact_7503_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_7504_ln__le__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_7505_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_7506_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self2
thf(fact_7507_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_7508_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_7509_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P2: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P2 @ I2 @ J2 ) ) )
       => ( P2 @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_7510_card__length__sum__list__rec,axiom,
    ! [M: nat,N8: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L4: list_nat] :
                ( ( ( size_size_list_nat @ L4 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L4 )
                  = N8 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L4: list_nat] :
                  ( ( ( size_size_list_nat @ L4 )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L4 )
                    = N8 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L4: list_nat] :
                  ( ( ( size_size_list_nat @ L4 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L4 ) @ one_one_nat )
                    = N8 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_7511_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
            & ( ord_less_int @ R @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R )
                & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_7512_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_7513_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R: int,Q7: int,R6: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q7 @ R6 ) )
       => ( Q3 = Q7 ) ) ) ).

% unique_quotient
thf(fact_7514_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R: int,Q7: int,R6: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q7 @ R6 ) )
       => ( R = R6 ) ) ) ).

% unique_remainder
thf(fact_7515_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_7516_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R ) ) ).

% mod_int_unique
thf(fact_7517_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R ) )
     => ( ( divide_divide_int @ K @ L )
        = Q3 ) ) ).

% div_int_unique
thf(fact_7518_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q3 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_7519_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_7520_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_7521_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [R4: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
          & ! [N9: nat] :
              ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S3 ) )
             => ( member_nat @ ( R4 @ N9 ) @ S3 ) ) ) ) ).

% finite_enumerate
thf(fact_7522_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_7523_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_7524_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A13: int,A24: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A13 = K3 )
              & ( A24 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L4: int,K3: int,Q8: int] :
              ( ( A13 = K3 )
              & ( A24 = L4 )
              & ( A32
                = ( product_Pair_int_int @ Q8 @ zero_zero_int ) )
              & ( L4 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q8 @ L4 ) ) )
          | ? [R5: int,L4: int,K3: int,Q8: int] :
              ( ( A13 = K3 )
              & ( A24 = L4 )
              & ( A32
                = ( product_Pair_int_int @ Q8 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L4 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L4 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q8 @ L4 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_7525_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q5: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q5 @ A23 ) ) ) )
         => ~ ! [R4: int,Q5: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q5 @ R4 ) )
               => ( ( ( sgn_sgn_int @ R4 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q5 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_7526_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_7527_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_7528_div__sgn__abs__cancel,axiom,
    ! [V2: int,K: int,L: int] :
      ( ( V2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V2 ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V2 ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_7529_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_7530_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% sinh_real_less_iff
thf(fact_7531_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% sinh_real_le_iff
thf(fact_7532_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% sinh_real_pos_iff
thf(fact_7533_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_7534_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_7535_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_7536_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_7537_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_7538_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A7: real] : ( if_real @ ( A7 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A7 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_7539_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_7540_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N6: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N6 ) @ ( modulo_modulo_nat @ M5 @ N6 ) ) ) ) ).

% divmod_nat_def
thf(fact_7541_ln__root,axiom,
    ! [N: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_7542_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_7543_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = ( root @ N @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_7544_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_7545_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_7546_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_7547_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_7548_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_7549_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_7550_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_7551_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_7552_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_7553_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_7554_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_7555_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_7556_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_7557_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_7558_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_7559_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_7560_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_7561_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_7562_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_7563_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_7564_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).

% real_root_abs
thf(fact_7565_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_7566_real__root__strict__decreasing,axiom,
    ! [N: nat,N8: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N8 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N8 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_7567_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_7568_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_7569_real__root__strict__increasing,axiom,
    ! [N: nat,N8: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N8 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N8 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_7570_real__root__decreasing,axiom,
    ! [N: nat,N8: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N8 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N8 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_7571_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_7572_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X )
         => ( ( root @ N @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_7573_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N @ ( power_power_real @ X @ N ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_7574_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
        = X ) ) ).

% sgn_power_root
thf(fact_7575_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_7576_real__root__increasing,axiom,
    ! [N: nat,N8: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N8 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N8 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_7577_split__root,axiom,
    ! [P2: real > $o,N: nat,X: real] :
      ( ( P2 @ ( root @ N @ X ) )
      = ( ( ( N = zero_zero_nat )
         => ( P2 @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y4: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N ) )
                = X )
             => ( P2 @ Y4 ) ) ) ) ) ).

% split_root
thf(fact_7578_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_7579_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_7580_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_7581_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
      = ( X != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_7582_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_7583_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_7584_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_real @ one_one_real @ X ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_7585_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_real @ X @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_7586_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_real @ A @ X ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_7587_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_real @ X @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_7588_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_7589_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_7590_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X )
          = one_one_real )
        = ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_7591_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_7592_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_7593_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_7594_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_7595_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_7596_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ A @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_7597_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_7598_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_7599_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y ) )
          = Y ) ) ) ).

% log_powr_cancel
thf(fact_7600_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( powr_real @ A @ ( log @ A @ X ) )
            = X ) ) ) ) ).

% powr_log_cancel
thf(fact_7601_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_7602_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% less_log_iff
thf(fact_7603_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_7604_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_7605_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_less_iff
thf(fact_7606_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_7607_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_7608_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_7609_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_7610_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_7611_powr__non__neg,axiom,
    ! [A: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_7612_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_7613_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_7614_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_7615_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_less_mono
thf(fact_7616_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X )
       => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_mono
thf(fact_7617_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_7618_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_7619_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_7620_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_7621_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_7622_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_7623_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_7624_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X )
            = ( powr_real @ A @ Y ) )
          = ( X = Y ) ) ) ) ).

% powr_inj
thf(fact_7625_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_7626_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_7627_powr__le1,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_7628_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_7629_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_7630_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X )
          = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_7631_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X @ N ) ) ) ).

% powr_realpow
thf(fact_7632_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_7633_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_7634_powr__neg__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X ) ) ) ).

% powr_neg_one
thf(fact_7635_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_7636_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_7637_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_7638_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_7639_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_7640_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_7641_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_7642_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_7643_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_7644_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( ( log @ A @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_7645_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_7646_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_7647_real__of__int__div4,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_7648_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N6: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N6 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_7649_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N6: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N6 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).

% int_less_real_le
thf(fact_7650_real__of__int__div2,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_7651_real__of__int__div3,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_7652_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_7653_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_7654_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_7655_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_7656_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_7657_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W2 @ Z ) ) ) ).

% zless_nat_conj
thf(fact_7658_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_7659_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_7660_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_7661_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_7662_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_7663_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_7664_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_7665_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_7666_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_7667_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_7668_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_7669_eq__nat__nat__iff,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z7 ) )
          = ( Z = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_7670_all__nat,axiom,
    ( ( ^ [P4: nat > $o] :
        ! [X7: nat] : ( P4 @ X7 ) )
    = ( ^ [P5: nat > $o] :
        ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).

% all_nat
thf(fact_7671_ex__nat,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X7: nat] : ( P4 @ X7 ) )
    = ( ^ [P5: nat > $o] :
        ? [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
          & ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).

% ex_nat
thf(fact_7672_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_7673_floor__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq
thf(fact_7674_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_7675_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_7676_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_7677_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_mono_iff
thf(fact_7678_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_7679_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_7680_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_7681_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_7682_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_7683_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq2
thf(fact_7684_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_7685_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_7686_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_7687_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ( nat2 @ W2 )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_7688_split__nat,axiom,
    ! [P2: nat > $o,I: int] :
      ( ( P2 @ ( nat2 @ I ) )
      = ( ! [N6: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N6 ) )
           => ( P2 @ N6 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P2 @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_7689_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W2 @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_7690_nat__add__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_7691_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_7692_Suc__as__int,axiom,
    ( suc
    = ( ^ [A7: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A7 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_7693_nat__mult__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_7694_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_7695_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_7696_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_7697_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_7698_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_7699_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_7700_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_7701_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_7702_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_7703_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_7704_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
        = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_7705_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_7706_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_7707_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_7708_diff__nat__eq__if,axiom,
    ! [Z7: int,Z: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_7709_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_7710_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X4: real] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X4 )
              & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_7711_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_7712_forall__pos__mono__1,axiom,
    ! [P2: real > $o,E2: real] :
      ( ! [D2: real,E: real] :
          ( ( ord_less_real @ D2 @ E )
         => ( ( P2 @ D2 )
           => ( P2 @ E ) ) )
     => ( ! [N2: nat] : ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P2 @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_7713_forall__pos__mono,axiom,
    ! [P2: real > $o,E2: real] :
      ( ! [D2: real,E: real] :
          ( ( ord_less_real @ D2 @ E )
         => ( ( P2 @ D2 )
           => ( P2 @ E ) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P2 @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_7714_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N6: nat] :
            ( ( N6 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N6 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N6 ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_7715_ln__inverse,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).

% ln_inverse
thf(fact_7716_log__inverse,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( log @ A @ ( inverse_inverse_real @ X ) )
            = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).

% log_inverse
thf(fact_7717_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_7718_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).

% Inf_nat_def1
thf(fact_7719_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).

% Sup_nat_def
thf(fact_7720_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X4: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_7721_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X4: rat] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 )
              & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_7722_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_rat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% less_eq_rat_def
thf(fact_7723_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A7: rat] : ( if_rat @ ( A7 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A7 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_7724_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A7: rat] : ( if_rat @ ( ord_less_rat @ A7 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A7 ) @ A7 ) ) ) ).

% abs_rat_def
thf(fact_7725_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R )
     => ~ ! [S4: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S4 )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R
                 != ( plus_plus_rat @ S4 @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_7726_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_7727_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P3 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P3 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_7728_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( ( ord_less_eq_nat @ M9 @ M5 )
         => ! [N6: nat] :
              ( ( ord_less_eq_nat @ M9 @ N6 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_7729_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_7730_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_7731_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).

% abs_sin_x_le_abs_x
thf(fact_7732_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_7733_sin__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).

% sin_ge_minus_one
thf(fact_7734_abs__sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_7735_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P3: int,Q3: int] :
      ( ( ( normalize @ R )
        = ( product_Pair_int_int @ P3 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_7736_mod__or__dist,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( sup_sup_assn @ P2 @ Q2 ) @ H2 )
      = ( ( rep_assn @ P2 @ H2 )
        | ( rep_assn @ Q2 @ H2 ) ) ) ).

% mod_or_dist
thf(fact_7737_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      ( ( sup_sup_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          | B ) ) ) ).

% merge_pure_or
thf(fact_7738_mod__h__bot__iff_I7_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        | ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_7739_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_7740_ent__disjE,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( ( entails @ A4 @ C3 )
     => ( ( entails @ B4 @ C3 )
       => ( entails @ ( sup_sup_assn @ A4 @ B4 ) @ C3 ) ) ) ).

% ent_disjE
thf(fact_7741_ent__disjI1,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( ( entails @ ( sup_sup_assn @ P2 @ Q2 ) @ R2 )
     => ( entails @ P2 @ R2 ) ) ).

% ent_disjI1
thf(fact_7742_ent__disjI2,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( ( entails @ ( sup_sup_assn @ P2 @ Q2 ) @ R2 )
     => ( entails @ Q2 @ R2 ) ) ).

% ent_disjI2
thf(fact_7743_ent__disjI1_H,axiom,
    ! [A4: assn,B4: assn,C3: assn] :
      ( ( entails @ A4 @ B4 )
     => ( entails @ A4 @ ( sup_sup_assn @ B4 @ C3 ) ) ) ).

% ent_disjI1'
thf(fact_7744_ent__disjI2_H,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( ( entails @ A4 @ C3 )
     => ( entails @ A4 @ ( sup_sup_assn @ B4 @ C3 ) ) ) ).

% ent_disjI2'
thf(fact_7745_ent__disjI1__direct,axiom,
    ! [A4: assn,B4: assn] : ( entails @ A4 @ ( sup_sup_assn @ A4 @ B4 ) ) ).

% ent_disjI1_direct
thf(fact_7746_ent__disjI2__direct,axiom,
    ! [B4: assn,A4: assn] : ( entails @ B4 @ ( sup_sup_assn @ A4 @ B4 ) ) ).

% ent_disjI2_direct
thf(fact_7747_star__or__dist1,axiom,
    ! [A4: assn,B4: assn,C3: assn] :
      ( ( times_times_assn @ ( sup_sup_assn @ A4 @ B4 ) @ C3 )
      = ( sup_sup_assn @ ( times_times_assn @ A4 @ C3 ) @ ( times_times_assn @ B4 @ C3 ) ) ) ).

% star_or_dist1
thf(fact_7748_star__or__dist2,axiom,
    ! [C3: assn,A4: assn,B4: assn] :
      ( ( times_times_assn @ C3 @ ( sup_sup_assn @ A4 @ B4 ) )
      = ( sup_sup_assn @ ( times_times_assn @ C3 @ A4 ) @ ( times_times_assn @ C3 @ B4 ) ) ) ).

% star_or_dist2
thf(fact_7749_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_7750_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_7751_mod__star__conv,axiom,
    ! [A4: assn,B4: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A4 @ B4 ) @ H2 )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As2: set_nat] :
            ( ( H2
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As2 ) ) )
            & ( ( inf_inf_set_nat @ As1 @ As2 )
              = bot_bot_set_nat )
            & ( rep_assn @ A4 @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
            & ( rep_assn @ B4 @ ( produc7507926704131184380et_nat @ Hr @ As2 ) ) ) ) ) ).

% mod_star_conv
thf(fact_7752_star__assnI,axiom,
    ! [P2: assn,H2: heap_e7401611519738050253t_unit,As3: set_nat,Q2: assn,As4: set_nat] :
      ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
     => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As4 ) )
       => ( ( ( inf_inf_set_nat @ As3 @ As4 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As3 @ As4 ) ) ) ) ) ) ).

% star_assnI
thf(fact_7753_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_7754_range__enumerate,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S3 ) @ top_top_set_nat )
        = S3 ) ) ).

% range_enumerate
thf(fact_7755_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_7756_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_7757_card__nat,axiom,
    ( ( finite_card_nat @ top_top_set_nat )
    = zero_zero_nat ) ).

% card_nat
thf(fact_7758_card__num0,axiom,
    ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
    = zero_zero_nat ) ).

% card_num0
thf(fact_7759_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_7760_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_7761_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      ( ( inf_inf_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_and
thf(fact_7762_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_7763_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_7764_mod__h__bot__iff_I6_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_7765_card__literal,axiom,
    ( ( finite_card_literal @ top_top_set_literal )
    = zero_zero_nat ) ).

% card_literal
thf(fact_7766_ent__conjE2,axiom,
    ! [B4: assn,C3: assn,A4: assn] :
      ( ( entails @ B4 @ C3 )
     => ( entails @ ( inf_inf_assn @ A4 @ B4 ) @ C3 ) ) ).

% ent_conjE2
thf(fact_7767_ent__conjE1,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( ( entails @ A4 @ C3 )
     => ( entails @ ( inf_inf_assn @ A4 @ B4 ) @ C3 ) ) ).

% ent_conjE1
thf(fact_7768_ent__conjI,axiom,
    ! [A4: assn,B4: assn,C3: assn] :
      ( ( entails @ A4 @ B4 )
     => ( ( entails @ A4 @ C3 )
       => ( entails @ A4 @ ( inf_inf_assn @ B4 @ C3 ) ) ) ) ).

% ent_conjI
thf(fact_7769_mod__and__dist,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ H2 )
      = ( ( rep_assn @ P2 @ H2 )
        & ( rep_assn @ Q2 @ H2 ) ) ) ).

% mod_and_dist
thf(fact_7770_ent__true,axiom,
    ! [P2: assn] : ( entails @ P2 @ top_top_assn ) ).

% ent_true
thf(fact_7771_mod__star__trueE,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H2 )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P2 @ H5 ) ) ).

% mod_star_trueE
thf(fact_7772_mod__star__trueI,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P2 @ H2 )
     => ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_7773_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_7774_root__def,axiom,
    ( root
    = ( ^ [N6: nat,X4: real] :
          ( if_real @ ( N6 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y4: real] : ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N6 ) )
            @ X4 ) ) ) ) ).

% root_def
thf(fact_7775_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A7: assn,B7: assn] :
          ( A7
          = ( inf_inf_assn @ A7 @ B7 ) ) ) ) ).

% less_eq_assn_def
thf(fact_7776_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A7: assn,B7: assn] : ( inf_inf_assn @ A7 @ ( uminus_uminus_assn @ B7 ) ) ) ) ).

% minus_assn_def
thf(fact_7777_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa2: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa2 @ Xb )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ? [As12: set_nat,As22: set_nat] :
                ( ( As
                  = ( sup_sup_set_nat @ As12 @ As22 ) )
                & ( ( inf_inf_set_nat @ As12 @ As22 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
                & ( Xa2 @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_7778_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A7: assn,B7: assn] :
          ( ( ord_less_eq_assn @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% less_assn_def
thf(fact_7779_binomial__def,axiom,
    ( binomial
    = ( ^ [N6: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K5: set_nat] :
                ( ( member_set_nat @ K5 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N6 ) ) )
                & ( ( finite_card_nat @ K5 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_7780_times__assn__raw_Osimps,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( times_assn_raw @ P2 @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
      = ( ? [As1: set_nat,As2: set_nat] :
            ( ( As3
              = ( sup_sup_set_nat @ As1 @ As2 ) )
            & ( ( inf_inf_set_nat @ As1 @ As2 )
              = bot_bot_set_nat )
            & ( P2 @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
            & ( Q2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_7781_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa2: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa2 @ Xb )
        = Y )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( Y
              = ( ~ ? [As1: set_nat,As2: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As1 @ As2 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As2 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
                      & ( Xa2 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_7782_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa2: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa2 @ Xb )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ~ ? [As13: set_nat,As23: set_nat] :
                  ( ( As
                    = ( sup_sup_set_nat @ As13 @ As23 ) )
                  & ( ( inf_inf_set_nat @ As13 @ As23 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
                  & ( Xa2 @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_7783_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa2: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa2 @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( Y
                  = ( ? [As1: set_nat,As2: set_nat] :
                        ( ( As
                          = ( sup_sup_set_nat @ As1 @ As2 ) )
                        & ( ( inf_inf_set_nat @ As1 @ As2 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
                        & ( Xa2 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa2 @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_7784_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa2: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa2 @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa2 @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa2 @ ( produc7507926704131184380et_nat @ H @ As ) ) ) )
               => ~ ? [As13: set_nat,As23: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As13 @ As23 ) )
                      & ( ( inf_inf_set_nat @ As13 @ As23 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
                      & ( Xa2 @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_7785_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa2: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa2 @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa2 @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa2 @ ( produc7507926704131184380et_nat @ H @ As ) ) ) )
               => ? [As12: set_nat,As22: set_nat] :
                    ( ( As
                      = ( sup_sup_set_nat @ As12 @ As22 ) )
                    & ( ( inf_inf_set_nat @ As12 @ As22 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
                    & ( Xa2 @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_7786_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_7787_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_7788_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa2 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_7789_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_7790_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_7791_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_7792_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_7793_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_7794_sorted__upto,axiom,
    ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).

% sorted_upto
thf(fact_7795_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_7796_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_7797_upto_Osimps,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_7798_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa2 )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa2 )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_7799_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_7800_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_7801_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_7802_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_7803_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_7804_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_7805_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_7806_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_7807_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_7808_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_7809_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_7810_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_7811_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_7812_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_7813_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_7814_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_7815_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_7816_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_7817_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_7818_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_7819_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_7820_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_7821_enat__less__induct,axiom,
    ! [P2: extended_enat > $o,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
             => ( P2 @ M2 ) )
         => ( P2 @ N2 ) )
     => ( P2 @ N ) ) ).

% enat_less_induct
thf(fact_7822_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_7823_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_7824_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_7825_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_7826_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_7827_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T @ X ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% insert'_correct
thf(fact_7828_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ S @ M )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
           => ( vEBT_invar_vebt @ X3 @ N ) )
       => ( ( M
            = ( suc @ N ) )
         => ( ! [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 @ N ) ) ) ) )
           => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
             => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_7829_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_7830_mulcomm,axiom,
    ! [I: nat,Va2: nat] :
      ( ( times_times_nat @ I @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ I ) ) ).

% mulcomm
thf(fact_7831_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% two_realpow_ge_two
thf(fact_7832_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_7833_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T @ X ) @ N ) ) ) ).

% valid_pres_insert
thf(fact_7834_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_7835_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_7836_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% two_powr_height_bound_deg
thf(fact_7837_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_7838_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_7839_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_7840_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_7841_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_7842_log__ceil__idem,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_7843_count__buildup,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% count_buildup
thf(fact_7844_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_7845_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_7846_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_7847_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_7848_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_7849_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_7850_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_7851_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: 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 ) @ X ) @ one_one_nat ) ) ).

% tdeletemimi'
thf(fact_7852_count__buildup_H,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% count_buildup'
thf(fact_7853_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_7854_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_7855_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_7856_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_7857_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( 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 ) ) @ N ) @ one_one_real ) ) ) ) ).

% cnt_bound'
thf(fact_7858_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) ) @ ( 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_7859_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H4: nat,L4: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L4 ) ) ) ).

% bit_concat_def
thf(fact_7860_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND_o @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_memberi
thf(fact_7861_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_7862_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_7863_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_7864_insert__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X ) ) ) ) ) ).

% insert_correct
thf(fact_7865_set__vebt__insert,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X ) )
          = ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% set_vebt_insert
thf(fact_7866_insert__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X ) ) ) ) ) ).

% insert_corr
thf(fact_7867_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_7868_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_7869_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_7870_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M3: nat] :
              ( ( ( some_nat @ M3 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_7871_add__2__eq__Suc_H,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc'
thf(fact_7872_add__2__eq__Suc,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc
thf(fact_7873_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_7874_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_7875_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_7876_zmod__numeral__Bit0,axiom,
    ! [V2: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_7877_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_7878_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_7879_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_7880_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_7881_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_7882_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_7883_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_7884_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_7885_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_7886_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_7887_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_7888_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_7889_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_7890_n__less__equal__power__2,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% n_less_equal_power_2
thf(fact_7891_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_7892_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_7893_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_7894_less__2__cases,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_7895_nat__induct2,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ( P2 @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( P2 @ N2 )
             => ( P2 @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct2
thf(fact_7896_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_7897_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_7898_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_7899_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_7900_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_7901_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_7902_not__exp__less__eq__0__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_7903_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_7904_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_7905_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_7906_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_7907_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero_02
thf(fact_7908_mod__exp__less__eq__exp,axiom,
    ! [A: int,N: nat] : ( ord_less_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% mod_exp_less_eq_exp
thf(fact_7909_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_7910_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_7911_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_7912_nat__bit__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P2 @ N2 )
           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( P2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
       => ( ! [N2: nat] :
              ( ( P2 @ N2 )
             => ( P2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_7913_two__pow__div__gt__le,axiom,
    ! [V2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ V2 @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% two_pow_div_gt_le
thf(fact_7914_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
      ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus_nat @ M @ N ) )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ Y ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Sz ) ) ) ) ) ).

% nat_add_offset_less
thf(fact_7915_nat__power__less__diff,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Q3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     => ( ord_less_nat @ Q3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% nat_power_less_diff
thf(fact_7916_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_7917_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( ( ord_less_eq_nat @ N @ ( 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 ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_le_power_trans
thf(fact_7918_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_7919_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_7920_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_7921_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_7922_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_7923_exp__plus__inverse__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_7924_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,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_7925_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N6: nat] : ( divide_divide_nat @ ( times_times_nat @ N6 @ ( suc @ N6 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_7926_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 )
       => ? [N2: 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 ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_7927_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 )
       => ? [N2: 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 ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_7928_less__two__pow__divD,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
        & ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_7929_less__two__pow__divI,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% less_two_pow_divI
thf(fact_7930_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( ( ord_less_nat @ N @ ( 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 ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_less_power_trans
thf(fact_7931_num_Osize_I5_J,axiom,
    ! [X2: num] :
      ( ( size_size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_7932_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C2: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C2 ) ) @ ( 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 @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_7933_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_7934_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_7935_Suc__nat__number__of__add,axiom,
    ! [V2: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_7936_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X: nat,Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
      & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% VEBT_internal.insert'.simps(2)
thf(fact_7937_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( divide_divide_nat @ N @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_div_eq_Suc_0_iff
thf(fact_7938_power__2__mult__step__le,axiom,
    ! [N3: nat,N: nat,K6: nat,K: nat] :
      ( ( ord_less_eq_nat @ N3 @ N )
     => ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K6 ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( plus_plus_nat @ K6 @ one_one_nat ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_7939_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_7940_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_7941_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_7942_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_7943_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_7944_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_7945_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_7946_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_7947_real__le__abs__sinh,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_7948_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q8: nat] : ( ord_less_nat @ Q8 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_7949_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_7950_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_7951_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ~ ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) )
                  & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
thf(fact_7952_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_7953_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_7954_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_7955_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_7956_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_7957_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_7958_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_7959_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_7960_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_7961_sinh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( sinh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_7962_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d2 @ T @ X ) ) @ ( 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_7963_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c2 @ T @ X ) ) @ ( 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_7964_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : X4
          @ ( set_or1266510415728281911st_int @ M @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_7965_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_7966_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_7967_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_7968_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_7969_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_7970_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_7971_binomial__code,axiom,
    ( binomial
    = ( ^ [N6: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N6 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N6 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N6 @ ( minus_minus_nat @ N6 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N6 @ K3 ) @ one_one_nat ) @ N6 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_7972_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_7973_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_7974_arctan__double,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_7975_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_7976_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_7977_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_7978_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_7979_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_7980_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_7981_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_7982_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_7983_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi
thf(fact_7984_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( vEBT_V9176841429113362141ildupi @ N ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi'
thf(fact_7985_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
        @ ^ [R5: $o] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_member @ T @ X ) ) ) )
        @ ( 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 @ N ) ) ) ) ) ) ) ) ).

% htt_vebt_memberi_invar_vebt
thf(fact_7986_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( minus_minus_nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi''
thf(fact_7987_htt__vebt__predi,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_pred @ T @ X ) ) ) )
        @ ( 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 @ N ) ) ) ) ) ) ) ) ).

% htt_vebt_predi
thf(fact_7988_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T2: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).

% Tb_Tb'
thf(fact_7989_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% T_vebt_buildupi_univ
thf(fact_7990_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: 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 ) @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_7991_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_7992_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_7993_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_7994_Tb_H__cnt,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N ) ) ) ) ).

% Tb'_cnt
thf(fact_7995_TBOUND__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_predi
thf(fact_7996_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_succi
thf(fact_7997_T__vebt__buildupi__cnt_H,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) ) ) ).

% T_vebt_buildupi_cnt'
thf(fact_7998_TBOUND__buildupi,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% TBOUND_buildupi
thf(fact_7999_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( 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_8000_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( 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 ) ) @ N ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cnt_bound
thf(fact_8001_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ ( 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_8002_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi_univ
thf(fact_8003_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi'_univ
thf(fact_8004_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) ) @ ( 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_8005_time__vebt__delete,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) ) @ ( 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 ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ) ).

% time_vebt_delete
thf(fact_8006_time__vebt__pred,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d @ T @ X ) ) @ ( 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 ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ) ).

% time_vebt_pred
thf(fact_8007_time__vebt__succ,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c @ T @ X ) ) @ ( 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 ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ) ).

% time_vebt_succ
thf(fact_8008_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) @ ( 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_8009_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_8010_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_8011_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_8012_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_8013_TBOUND__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V1365221501068881998eletei @ T @ Ti @ X ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) ) @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ) ).

% TBOUND_vebt_deletei
thf(fact_8014_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_member @ T @ X ) ) ) )
      @ ( 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_8015_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_8016_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_8017_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_8018_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) @ ( 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 @ N ) ) ) ) ) ) ) ) ).

% htt_vebt_inserti_invar_vebt
thf(fact_8019_htt__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ T @ X ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ) ) ).

% htt_vebt_deletei
thf(fact_8020_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_succ @ T @ X ) ) ) )
        @ ( 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 @ N ) ) ) ) ) ) ) ) ).

% htt_vebt_succi
thf(fact_8021_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V2: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V2 ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V2 ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_8022_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_8023_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V2: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V2 ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V2 ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_8024_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_8025_zmod__numeral__Bit1,axiom,
    ! [V2: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_8026_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_8027_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_8028_eval__nat__numeral_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_8029_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_8030_Suc3__eq__add__3,axiom,
    ! [N: nat] :
      ( ( suc @ ( suc @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).

% Suc3_eq_add_3
thf(fact_8031_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_8032_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_8033_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_8034_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_8035_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_8036_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_8037_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_8038_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_8039_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_8040_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_8041_insertsimp,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( 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_8042_small__powers__of__2,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ X )
     => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ X @ one_one_nat ) ) ) ) ).

% small_powers_of_2
thf(fact_8043_log__base__10__eq2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq2
thf(fact_8044_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d @ T @ X ) @ ( 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_8045_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) @ ( 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_8046_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c @ T @ X ) @ ( 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_8047_log__base__10__eq1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq1
thf(fact_8048_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d @ T @ X ) ) @ ( 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_8049_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c @ T @ X ) ) @ ( 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_8050_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) ) @ ( 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_8051_VEBT__internal_Ovebt__buildupi__rule,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% VEBT_internal.vebt_buildupi_rule
thf(fact_8052_vebt__buildupi__rule,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ bot_bot_set_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% vebt_buildupi_rule
thf(fact_8053_vebt__memberi__rule,axiom,
    ! [N: nat,S: set_nat,Ti: vEBT_VEBTi,X: nat] :
      ( time_htt_o @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( member_nat @ X @ S ) ) ) )
      @ ( 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 @ N ) ) ) ) ) ) ) ).

% vebt_memberi_rule
thf(fact_8054_vebt__predi__rule,axiom,
    ! [N: nat,S: set_nat,Ti: vEBT_VEBTi,X: nat,Y: nat] :
      ( time_htt_option_nat @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
          @ ( pure_assn
            @ ( ( R5
                = ( some_nat @ Y ) )
              = ( vEBT_is_pred_in_set @ S @ X @ Y ) ) ) )
      @ ( 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 @ N ) ) ) ) ) ) ) ).

% vebt_predi_rule
thf(fact_8055_vebt__succi__rule,axiom,
    ! [N: nat,S: set_nat,Ti: vEBT_VEBTi,X: nat,Y: nat] :
      ( time_htt_option_nat @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
          @ ( pure_assn
            @ ( ( R5
                = ( some_nat @ Y ) )
              = ( vEBT_is_succ_in_set @ S @ X @ Y ) ) ) )
      @ ( 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 @ N ) ) ) ) ) ) ) ).

% vebt_succi_rule
thf(fact_8056_vebt__inserti__rule,axiom,
    ! [X: nat,N: nat,S: set_nat,Ti: vEBT_VEBTi] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( sup_sup_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( 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 @ N ) ) ) ) ) ) ) ) ).

% vebt_inserti_rule
thf(fact_8057_space__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space_bound
thf(fact_8058_vebt__space__linear__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% vebt_space_linear_bound
thf(fact_8059_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( 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 ) ) @ N ) @ one_one_real ) ) ) ) ).

% space_2_pow_bound
thf(fact_8060_space__space_H,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).

% space_space'
thf(fact_8061_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space'_bound
thf(fact_8062_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_8063_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_8064_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_8065_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_8066_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( 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_8067_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_8068_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( 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_8069_t__build__cnt,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_build_cnt
thf(fact_8070_t__buildup__cnt,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8346862874174094_d_u_p @ N ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_buildup_cnt
thf(fact_8071_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ord_less_eq_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ U ) ) ) ).

% vebt_buildup_bound
thf(fact_8072_buildup__build__time,axiom,
    ! [N: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).

% buildup_build_time
thf(fact_8073_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_8074_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_8075_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_8076_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_8077_div__half__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( product_Pair_nat_nat @ ( divide_divide_nat @ X @ Y ) @ ( modulo_modulo_nat @ X @ Y ) )
        = ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ Y @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).

% div_half_nat
thf(fact_8078_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( 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_8079_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( 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_8080_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) )
                    & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
thf(fact_8081_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_8082_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X4: nat,N6: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ).

% high_def
thf(fact_8083_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_8084_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_8085_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_8086_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_8087_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_8088_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_8089_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X4: code_integer] : ( plus_p5714425477246183910nteger @ X4 @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_8090_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_8091_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_8092_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_8093_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_8094_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_8095_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_8096_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_8097_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_8098_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).

% abs_integer_code
thf(fact_8099_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_8100_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_8101_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va2 ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_8102_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_8103_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_8104_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_8105_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X4: nat,N6: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ).

% low_def
thf(fact_8106_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = X ) ) ).

% low_inv
thf(fact_8107_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_8108_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_8109_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_8110_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_8111_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_8112_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_8113_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_8114_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: 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 @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = 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 @ X @ ( 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 ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_8115_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_8116_del__x__not__mi,axiom,
    ! [Mi: nat,X: 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 @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_8117_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: 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 @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = 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 @ X @ ( 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 ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_8118_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 ) @ X )
                  = ( 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 @ ( X = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ 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 @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ 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_8119_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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_8120_del__x__mi__lets__in__minNull,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_8121_del__x__mi__lets__in,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_8122_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 ) @ X )
                    = ( 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_8123_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_8124_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_8125_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_8126_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_8127_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_8128_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_8129_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,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = one_one_nat ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_8130_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_8131_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_8132_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_8133_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_8134_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                @ ( if_nat
                  @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                     != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                    @ ( if_nat
                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                        = none_nat )
                      @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_8135_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                  @ ( if_nat
                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                       != none_nat )
                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                      @ ( if_nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          = none_nat )
                        @ one_one_nat
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_8136_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B3 ) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [N2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N2 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A3 @ B3 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_8137_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,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa2 )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa2
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ? [N2: nat] :
                    ( Xa2
                    = ( suc @ ( suc @ N2 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y = one_one_nat ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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'.elims
thf(fact_8138_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_8139_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa2
                  = ( suc @ N2 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_8140_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va ) ) )
                 => ~ ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( ( A3
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_8141_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_8142_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_8143_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ Uw ) )
           => ( ( Xa2
                = ( suc @ zero_zero_nat ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va ) ) )
                 => ( Y
                   != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_8144_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ B3 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa2
                  = ( suc @ N2 ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( Y
                       != ( plus_plus_nat @ one_one_nat
                          @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                              @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_8145_vebt__member__code_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
      = ( ~ ( ( Deg = zero_zero_nat )
            | ( Deg
              = ( suc @ zero_zero_nat ) ) )
        & ( ~ ( ( Deg = zero_zero_nat )
              | ( Deg
                = ( suc @ zero_zero_nat ) ) )
         => ( ( X != Mi )
           => ( ( X != Ma )
             => ( ~ ( ord_less_nat @ X @ Mi )
                & ( ~ ( ord_less_nat @ X @ Mi )
                 => ( ~ ( ord_less_nat @ Ma @ X )
                    & ( ~ ( ord_less_nat @ Ma @ X )
                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member_code(3)
thf(fact_8146_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
          @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
            @ ( if_nat
              @ ( ( X = Mi )
                & ( X = Ma ) )
              @ ( numeral_numeral_nat @ ( bit1 @ one ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( X = 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                            @ ( if_nat
                              @ ( ( ( X = Mi )
                                 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                    = Ma ) )
                                & ( ( X != Mi )
                                 => ( X = 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                @ ( plus_plus_nat @ one_one_nat
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      = none_nat )
                                    @ one_one_nat
                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ one_one_nat ) ) )
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                          @ ( if_nat
                            @ ( ( ( X = Mi )
                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                  = Ma ) )
                              & ( ( X != Mi )
                               => ( X = 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                            @ one_one_nat ) ) ) ) )
                  @ one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_8147_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
           => Y )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S4: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_8148_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S4: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_8149_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S4: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_8150_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
         => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_8151_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,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa2 )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa2
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ? [N2: nat] :
                    ( Xa2
                    = ( suc @ ( suc @ N2 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                              @ one_one_nat
                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                  @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                          @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ one_one_nat ) ) )
                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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.elims
thf(fact_8152_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: 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 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( 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_8153_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_8154_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu: $o,Uv: $o] :
            ( X
           != ( vEBT_Leaf @ Uu @ Uv ) )
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
             => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_8155_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => Y )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_8156_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( X = Mi ) @ zero_zero_nat
          @ ( if_nat @ ( X = Ma ) @ zero_zero_nat
            @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ zero_zero_nat
              @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ zero_zero_nat
                @ ( if_nat
                  @ ( ( ord_less_nat @ Mi @ X )
                    & ( ord_less_nat @ X @ Ma ) )
                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                  @ zero_zero_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_8157_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_8158_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_8159_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: 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 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( 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_8160_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
         => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
           => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_8161_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => Y )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
             => Y )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_8162_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa2 )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ one_one_nat
                        @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
                          @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
                            @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Mi2 @ Xa2 )
                                    & ( ord_less_nat @ Xa2 @ Ma2 ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( 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'.elims
thf(fact_8163_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ~ ( ( X = Mi )
                | ( X = Ma ) ) )
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
          @ one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_8164_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa2 )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( if_nat
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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'.elims
thf(fact_8165_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A13: vEBT_VEBT,A24: nat] :
          ( ( ? [A7: $o,B7: $o] :
                ( A13
                = ( vEBT_Leaf @ A7 @ B7 ) )
            & ( A24
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
              ( ( A13
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A24 @ TreeList4 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N6 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N6 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) )
              & ( A24
                = ( plus_plus_nat @ N6 @ N6 ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
              ( ( A13
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A24 @ TreeList4 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N6 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N6 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N6 ) ) )
              & ( A24
                = ( plus_plus_nat @ N6 @ ( suc @ N6 ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A13
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N6 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N6 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) )
              & ( A24
                = ( plus_plus_nat @ N6 @ N6 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A24 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N6 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N6 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N6 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N6 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A13
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N6 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N6 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N6 ) ) )
              & ( A24
                = ( plus_plus_nat @ N6 @ ( suc @ N6 ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N6 ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A24 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N6 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N6 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N6 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N6 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N6 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_8166_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A3: $o,B3: $o] :
              ( A12
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X5 @ N2 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N2 @ M3 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X5 @ N2 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                       => ( ( M3
                            = ( suc @ N2 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N2 @ M3 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X5 @ N2 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N2 @ M3 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                & ! [X5: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X5 )
                                                      & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X5 @ N2 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                           => ( ( M3
                                = ( suc @ N2 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N2 @ M3 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                  & ! [X5: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X5 )
                                                        & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_8167_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) ) )
           => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_8168_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_8169_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_8170_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa2 )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                        @ ( if_nat
                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                            & ~ ( ( Xa2 = Mi2 )
                                | ( Xa2 = 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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.elims
thf(fact_8171_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ Uw ) )
           => ( ( Xa2
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ? [Va: nat] :
                    ( Xa2
                    = ( suc @ ( suc @ Va ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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'.elims
thf(fact_8172_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ B3 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa2
                  = ( suc @ N2 ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y = one_one_nat ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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'.elims
thf(fact_8173_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N6: nat,TreeList4: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X4 @ N6 ) ) @ ( vEBT_VEBT_low @ X4 @ N6 ) ) ) ) ).

% in_children_def
thf(fact_8174_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,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( Y = 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 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y = 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 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                                  @ one_one_nat
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                      @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                              @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                    @ ( if_nat
                                                      @ ( ( ( Xa2 = Mi2 )
                                                         => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa2 != Mi2 )
                                                         => ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                      @ one_one_nat ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                  @ ( if_nat
                                                    @ ( ( ( Xa2 = Mi2 )
                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                          = Ma2 ) )
                                                      & ( ( Xa2 != Mi2 )
                                                       => ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_8175_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa2
                      = ( suc @ N2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y
                            = ( plus_plus_nat @ one_one_nat
                              @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat
                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_8176_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa2
                      = ( suc @ N2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_8177_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ 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 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Xa2 ) ) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ one_one_nat
                                @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_8178_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( ( B3
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B3
                           => ( ( A3
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A3
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Xa2 ) ) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_8179_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,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( Y = 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 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y = 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 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y = one_one_nat ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_8180_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A3 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A3 @ B3 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( 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 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_8181_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa2
                      = ( suc @ N2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_8182_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) @ Xa2 ) ) ) )
             => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                & ~ ( ( Xa2 = Mi2 )
                                    | ( Xa2 = 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_8183_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) @ Xa2 ) ) ) )
             => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_8184_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Xa2 ) ) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_8185_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S4 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S4 ) @ Xa2 ) ) ) )
             => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( if_nat
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_8186_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
                                @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
                                  @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
                                    @ ( if_nat
                                      @ ( ( ord_less_nat @ Mi2 @ Xa2 )
                                        & ( ord_less_nat @ Xa2 @ Ma2 ) )
                                      @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_8187_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa2 )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
             => ( Y
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( 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.elims
thf(fact_8188_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_8189_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,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_8190_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,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_8191_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_8192_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_8193_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% member_bound_size_univ
thf(fact_8194_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_8195_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
                 => ( ( Y
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_8196_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_8197_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_8198_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_8199_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa2 ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_8200_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_8201_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S4: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_8202_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( 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 @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_8203_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
               => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_8204_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
           => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_8205_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8206_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8207_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8208_sin__cos__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% sin_cos_npi
thf(fact_8209_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_8210_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_8211_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_8212_sin__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero
thf(fact_8213_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_8214_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8215_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8216_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ( ( sin_real @ X )
            = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_8217_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_8218_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_8219_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_8220_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_8221_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_8222_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_8223_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_8224_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8225_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_8226_sin__gt__zero2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero2
thf(fact_8227_sin__lt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_8228_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X )
                = ( sin_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_8229_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_8230_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_8231_arctan__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_8232_arctan__lbound,axiom,
    ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).

% arctan_lbound
thf(fact_8233_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8234_sin__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_8235_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_8236_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_8237_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X3 )
              = Y )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y5 )
                    = Y ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_8238_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8239_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8240_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8241_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N6: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_8242_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_8243_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_8244_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_8245_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_8246_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_8247_cos__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).

% cos_ge_minus_one
thf(fact_8248_abs__cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_8249_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_8250_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_8251_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_8252_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8253_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 )
      & ! [Y5: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
            & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y5 )
              = zero_zero_real ) )
         => ( Y5 = X3 ) ) ) ).

% cos_is_zero
thf(fact_8254_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_8255_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_8256_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ pi )
                  & ( ( cos_real @ Y5 )
                    = Y ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_8257_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_8258_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_8259_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_8260_cos__double__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_8261_cos__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero
thf(fact_8262_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_8263_cos__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_8264_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X4 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_8265_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
            & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X4 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_8266_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ pi )
            & ( X
              = ( cos_real @ T6 ) )
            & ( Y
              = ( sin_real @ T6 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_8267_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_8268_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_8269_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T6: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T6 ) )
              & ( Y
                = ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_8270_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T6 ) )
          & ( Y
            = ( sin_real @ T6 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_8271_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T6 ) )
               => ( Y
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8272_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8273_lowi__hT,axiom,
    ! [X: nat,N: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N ) ) )
      @ one_one_nat ) ).

% lowi_hT
thf(fact_8274_highi__hT,axiom,
    ! [X: nat,N: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N ) ) )
      @ one_one_nat ) ).

% highi_hT
thf(fact_8275_TBOUND__highi,axiom,
    ! [X: nat,N: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_highi @ X @ N ) @ one_one_nat ) ).

% TBOUND_highi
thf(fact_8276_TBOUND__lowi,axiom,
    ! [X: nat,N: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_lowi @ X @ N ) @ one_one_nat ) ).

% TBOUND_lowi
thf(fact_8277_highi__h,axiom,
    ! [X: nat,N: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N ) ) ) ) ).

% highi_h
thf(fact_8278_lowi__h,axiom,
    ! [X: nat,N: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N ) ) ) ) ).

% lowi_h
thf(fact_8279_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_8280_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_8281_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).

% set_bit_greater_eq
thf(fact_8282_set__encode__insert,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ N @ A4 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).

% set_encode_insert
thf(fact_8283_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N6: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N6 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N6 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8284_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N6: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N6 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N6 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N6 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_8285_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_8286_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_8287_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_8288_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_8289_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_8290_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_8291_even__Suc,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% even_Suc
thf(fact_8292_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_Suc_Suc_iff
thf(fact_8293_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_8294_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_8295_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% odd_Suc_minus_one
thf(fact_8296_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% even_diff_nat
thf(fact_8297_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_8298_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_8299_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_8300_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_8301_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_8302_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_8303_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_8304_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M @ N )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_8305_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_8306_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
      = ( ( ord_less_nat @ N @ M )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_minus_self
thf(fact_8307_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_8308_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_8309_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( dvd_dvd_nat @ M @ N )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_8310_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( nat_set_encode @ A4 )
            = ( nat_set_encode @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% set_encode_eq
thf(fact_8311_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).

% even_set_encode_iff
thf(fact_8312_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K @ N ) ) ) ).

% dvd_imp_le
thf(fact_8313_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_8314_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_8315_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D2 @ A )
          & ( dvd_dvd_nat @ D2 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_8316_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_8317_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q3 )
          = ( modulo_modulo_nat @ N @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_8318_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_8319_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_8320_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M5: nat,N6: nat] :
        ( ( dvd_dvd_nat @ M5 @ N6 )
        & ( M5 != N6 ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_8321_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( nat_set_encode @ A4 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_8322_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_8323_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_8324_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_8325_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_8326_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_8327_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R @ N )
     => ( ( ord_less_eq_nat @ R @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R ) ) ) ) ).

% mod_nat_eqI
thf(fact_8328_odd__pos,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% odd_pos
thf(fact_8329_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_8330_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_8331_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M5: nat] :
                      ( collect_nat
                      @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M5 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_8332_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_8333_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_8334_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_8335_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_8336_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_8337_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( suc @ zero_zero_nat ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( suc @ zero_zero_nat ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( 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 )
                     => ( Y
                        = ( 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.elims
thf(fact_8338_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
thf(fact_8339_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( 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 )
                     => ( Y
                        = ( 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'.elims
thf(fact_8340_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8341_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8342_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.simps(3)
thf(fact_8343_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_8344_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_8345_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_8346_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( 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 ) ) )
                     => ( Y
                        = ( 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.elims
thf(fact_8347_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( 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 )
                     => ( Y
                        = ( 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.elims
thf(fact_8348_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,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( 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 ) ) )
                     => ( Y
                        = ( 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.elims
thf(fact_8349_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,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( 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 ) ) )
                     => ( Y
                        = ( 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.elims
thf(fact_8350_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y != one_one_int ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( 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 )
                     => ( Y
                        = ( 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'.elims
thf(fact_8351_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_8352_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V3352910403632780892pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y = one_one_int )
               => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( 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 )
                       => ( Y
                          = ( 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 ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_8353_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,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
             => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( 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 ) ) )
                       => ( Y
                          = ( 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 ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_8354_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,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( 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 ) ) )
                       => ( Y
                          = ( 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 ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_8355_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M @ N )
         => ( ( dvd_dvd_int @ N @ M )
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_8356_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N )
       => ~ ( dvd_dvd_int @ N @ M ) ) ) ).

% zdvd_not_zless
thf(fact_8357_abs__div,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_8358_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( ( dvd_dvd_int @ Z @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z @ N ) ) ) ).

% zdvd_imp_le
thf(fact_8359_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_8360_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_8361_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ one_one_int @ M )
     => ( ( ( dvd_dvd_int @ M @ N )
         => ( ( divide_divide_int @ ( minus_minus_int @ N @ one_one_int ) @ M )
            = ( minus_minus_int @ ( divide_divide_int @ N @ M ) @ one_one_int ) ) )
        & ( ~ ( dvd_dvd_int @ M @ N )
         => ( ( divide_divide_int @ ( minus_minus_int @ N @ one_one_int ) @ M )
            = ( divide_divide_int @ N @ M ) ) ) ) ) ).

% int_div_sub_1
thf(fact_8362_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_8363_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_8364_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_8365_eme1p,axiom,
    ! [N: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
     => ( ( 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 @ N ) @ D )
            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ N @ D ) ) ) ) ) ) ).

% eme1p
thf(fact_8366_emep1,axiom,
    ! [N: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ D )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ N @ one_one_int ) @ D )
            = ( plus_plus_int @ ( modulo_modulo_int @ N @ D ) @ one_one_int ) ) ) ) ) ).

% emep1
thf(fact_8367_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( 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 ) ) )
                       => ( Y
                          = ( 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 ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_8368_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel2 @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_int @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_int @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( 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 )
                       => ( Y
                          = ( 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 ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
thf(fact_8369_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( 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 )
                       => ( Y
                          = ( 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 ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
thf(fact_8370_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V2957053500504383685pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( suc @ zero_zero_nat ) )
             => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( suc @ zero_zero_nat ) )
               => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( 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 )
                       => ( Y
                          = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_8371_vebt__delete__code_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
      = ( case_o2442805151034396888at_nat @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary )
        @ ^ [Mima: product_prod_nat_nat] :
            ( if_VEBT_VEBT @ ( ord_less_eq_nat @ Deg @ one_one_nat ) @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary )
            @ ( produc3169358591047799142T_VEBT
              @ ^ [Mi3: nat,Ma3: nat] :
                  ( if_VEBT_VEBT
                  @ ( ( ord_less_nat @ X @ Mi3 )
                    | ( ord_less_nat @ Ma3 @ X ) )
                  @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary )
                  @ ( if_VEBT_VEBT
                    @ ( ( X = Mi3 )
                      & ( X = Ma3 ) )
                    @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary )
                    @ ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( vEBT_Node
                          @ ( some_P7363390416028606310at_nat
                            @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi3 ) @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ Mi3 )
                              @ ( if_nat
                                @ ( ( ( X = Mi3 )
                                   => ( ( 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 ) ) ) ) ) )
                                      = Ma3 ) )
                                  & ( ( X != Mi3 )
                                   => ( X = Ma3 ) ) )
                                @ ( if_nat
                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    = none_nat )
                                  @ ( if_nat @ ( X = Mi3 ) @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ Mi3 )
                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                @ Ma3 ) ) )
                          @ Deg
                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( vEBT_Node
                          @ ( some_P7363390416028606310at_nat
                            @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi3 ) @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ Mi3 )
                              @ ( if_nat
                                @ ( ( ( X = Mi3 )
                                   => ( ( 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 ) ) ) ) ) )
                                      = Ma3 ) )
                                  & ( ( X != Mi3 )
                                   => ( X = Ma3 ) ) )
                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                @ Ma3 ) ) )
                          @ Deg
                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi3 ) @ ( 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ Summary ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary ) ) ) )
              @ Mima ) )
        @ Info ) ) ).

% vebt_delete_code(2)
thf(fact_8372_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N6: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N6 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N6 ) ) ) ) ) ).

% divmod_int_def
thf(fact_8373_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N6: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N6 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N6 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_8374_vebt__insert__code_I2_J,axiom,
    ! [Deg: nat,Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ( ord_less_eq_nat @ Deg @ one_one_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
      & ( ~ ( ord_less_eq_nat @ Deg @ one_one_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = ( case_o2442805151034396888at_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg @ TreeList @ Summary )
            @ ( produc3169358591047799142T_VEBT
              @ ^ [Mi3: nat,Ma3: nat] :
                  ( if_VEBT_VEBT
                  @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                    & ~ ( ( X = Mi3 )
                        | ( X = Ma3 ) ) )
                  @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ X @ Mi3 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ Mi3 @ X ) @ Ma3 ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
                  @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary ) ) )
            @ Info ) ) ) ) ).

% vebt_insert_code(2)
thf(fact_8375_minus__one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_div_numeral
thf(fact_8376_one__div__minus__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% one_div_minus_numeral
thf(fact_8377_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_8378_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_8379_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N6: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N6 = zero_zero_nat )
            | ( ord_less_nat @ M5 @ N6 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q8: nat] : ( product_Pair_nat_nat @ ( suc @ Q8 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N6 ) @ N6 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_8380_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L4: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q8: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L4 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L4 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_8381_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L4: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q8: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L4 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L4 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_8382_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L4: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q8: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L4 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L4 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_8383_vebt__succ__code_I2_J,axiom,
    ! [Deg: nat,Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ( ord_less_eq_nat @ Deg @ one_one_nat )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = none_nat ) )
      & ( ~ ( ord_less_eq_nat @ Deg @ one_one_nat )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = ( case_o1383228350324149268at_nat @ none_nat
            @ ( produc2484365769952853102on_nat
              @ ^ [Mi3: nat,Ma3: nat] :
                  ( if_option_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( some_nat @ Mi3 )
                  @ ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                    @ ( if_option_nat
                      @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                         != none_nat )
                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      @ ( if_option_nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          = none_nat )
                        @ none_nat
                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                    @ none_nat ) ) )
            @ Info ) ) ) ) ).

% vebt_succ_code(2)
thf(fact_8384_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_8385_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_18347121197199848620nteger ).

% integer_of_int_eq_of_int
thf(fact_8386_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N6: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N6 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N6 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_8387_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_8388_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_int @ Xa2 @ X ) ) ).

% less_integer.abs_eq
thf(fact_8389_divide__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa2 @ X ) ) ) ).

% divide_integer.abs_eq
thf(fact_8390_modulo__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( modulo_modulo_int @ Xa2 @ X ) ) ) ).

% modulo_integer.abs_eq
thf(fact_8391_integer__of__int__inject,axiom,
    ! [X: int,Y: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y @ top_top_set_int )
       => ( ( ( code_integer_of_int @ X )
            = ( code_integer_of_int @ Y ) )
          = ( X = Y ) ) ) ) ).

% integer_of_int_inject
thf(fact_8392_integer__of__int__induct,axiom,
    ! [P2: code_integer > $o,X: code_integer] :
      ( ! [Y3: int] :
          ( ( member_int @ Y3 @ top_top_set_int )
         => ( P2 @ ( code_integer_of_int @ Y3 ) ) )
     => ( P2 @ X ) ) ).

% integer_of_int_induct
thf(fact_8393_integer__of__int__cases,axiom,
    ! [X: code_integer] :
      ~ ! [Y3: int] :
          ( ( X
            = ( code_integer_of_int @ Y3 ) )
         => ~ ( member_int @ Y3 @ top_top_set_int ) ) ).

% integer_of_int_cases
thf(fact_8394_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_8395_Code__Numeral_Oset__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] :
      ( ( bit_se2793503036327961859nteger @ Xa2 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se7879613467334960850it_int @ Xa2 @ X ) ) ) ).

% Code_Numeral.set_bit_integer.abs_eq
thf(fact_8396_uminus__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( uminus1351360451143612070nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( uminus_uminus_int @ X ) ) ) ).

% uminus_integer.abs_eq
thf(fact_8397_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_eq_int @ Xa2 @ X ) ) ).

% less_eq_integer.abs_eq
thf(fact_8398_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa2 @ X ) ) ) ).

% plus_integer.abs_eq
thf(fact_8399_minus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X ) ) ) ).

% minus_integer.abs_eq
thf(fact_8400_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_8401_abs__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( abs_abs_Code_integer @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( abs_abs_int @ X ) ) ) ).

% abs_integer.abs_eq
thf(fact_8402_sgn__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( sgn_sgn_Code_integer @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( sgn_sgn_int @ X ) ) ) ).

% sgn_integer.abs_eq
thf(fact_8403_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_8404_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_8405_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_8406_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_8407_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_8408_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_8409_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_8410_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_8411_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_8412_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_8413_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_8414_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_8415_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_8416_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_8417_lowi__def,axiom,
    ( vEBT_VEBT_lowi
    = ( ^ [X4: nat,N6: nat] : ( heap_Time_return_nat @ ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ).

% lowi_def
thf(fact_8418_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_8419_highi__def,axiom,
    ( vEBT_VEBT_highi
    = ( ^ [X4: nat,N6: nat] : ( heap_Time_return_nat @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ).

% highi_def
thf(fact_8420_highsimp,axiom,
    ! [X: nat,N: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_high @ X @ N ) )
      = ( vEBT_VEBT_highi @ X @ N ) ) ).

% highsimp
thf(fact_8421_lowsimp,axiom,
    ! [X: nat,N: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_low @ X @ N ) )
      = ( vEBT_VEBT_lowi @ X @ N ) ) ).

% lowsimp
thf(fact_8422_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N10: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N10 @ N2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N6: nat] : ( abs_abs_real @ ( F @ N6 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_8423_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N6: nat] : ( abs_abs_real @ ( F @ N6 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N6: nat] : ( abs_abs_real @ ( F @ N6 ) ) ) ) ) ).

% summable_rabs
thf(fact_8424_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8425_suminf__eq__SUP__real,axiom,
    ! [X6: nat > real] :
      ( ( summable_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X6 @ I2 ) )
       => ( ( suminf_real @ X6 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X6 @ ( set_ord_lessThan_nat @ I3 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_8426_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8427_tanh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( tanh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_8428_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_8429_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_8430_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_8431_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_8432_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% tanh_real_less_iff
thf(fact_8433_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% tanh_real_le_iff
thf(fact_8434_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_8435_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% tanh_real_pos_iff
thf(fact_8436_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_8437_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_8438_flip__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] :
      ( ( bit_se1345352211410354436nteger @ Xa2 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se2159334234014336723it_int @ Xa2 @ X ) ) ) ).

% flip_bit_integer.abs_eq
thf(fact_8439_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] :
      ( ( bit_se8260200283734997820nteger @ Xa2 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se4203085406695923979it_int @ Xa2 @ X ) ) ) ).

% unset_bit_integer.abs_eq
thf(fact_8440_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_8441_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_8442_tanh__real__gt__neg1,axiom,
    ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).

% tanh_real_gt_neg1
thf(fact_8443_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8444_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_8445_tan__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_gt_zero
thf(fact_8446_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_8447_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_8448_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_8449_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).

% tan_monotone
thf(fact_8450_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y )
      & ! [Y5: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
            & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y5 )
              = Y ) )
         => ( Y5 = X3 ) ) ) ).

% tan_total
thf(fact_8451_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X3 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_8452_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_8453_tan__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_8454_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_8455_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_8456_tan__bound__pi2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_8457_arctan,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y ) )
        = Y ) ) ).

% arctan
thf(fact_8458_arctan__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X ) )
          = X ) ) ) ).

% arctan_tan
thf(fact_8459_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X )
            = Y )
         => ( ( arctan @ Y )
            = X ) ) ) ) ).

% arctan_unique
thf(fact_8460_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ 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 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_8461_arctan__def,axiom,
    ( arctan
    = ( ^ [Y4: real] :
          ( the_real
          @ ^ [X4: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
              & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X4 )
                = Y4 ) ) ) ) ) ).

% arctan_def
thf(fact_8462_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y4: real] :
          ( the_real
          @ ^ [X4: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
              & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X4 )
                = Y4 ) ) ) ) ) ).

% arcsin_def
thf(fact_8463_sin__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X )
        = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_8464_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% real_sqrt_less_iff
thf(fact_8465_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_8466_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_gt_0_iff
thf(fact_8467_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_8468_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_8469_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_8470_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_8471_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ one_one_real @ Y ) ) ).

% real_sqrt_gt_1_iff
thf(fact_8472_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_8473_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_8474_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_8475_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_8476_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_8477_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_8478_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_less_mono
thf(fact_8479_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_one
thf(fact_8480_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_8481_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_8482_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_gt_zero
thf(fact_8483_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_8484_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_8485_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_8486_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_8487_arcsin__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X ) )
          = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).

% arcsin_minus
thf(fact_8488_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_8489_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X )
            = ( arcsin @ Y ) )
          = ( X = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_8490_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8491_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_8492_cos__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_8493_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_8494_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_8495_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
     => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_8496_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_8497_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_8498_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_8499_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_8500_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_8501_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_8502_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_8503_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C2: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C2 ) @ ( 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 @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_8504_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_8505_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_8506_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_8507_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_8508_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_8509_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_8510_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_8511_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_8512_ln__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( sqrt @ X ) )
        = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_8513_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N )
          = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8514_arsinh__real__aux,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_8515_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8516_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8517_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_8518_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_8519_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8520_arcosh__real__def,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( arcosh_real @ X )
        = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_8521_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_8522_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_8523_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_8524_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_8525_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8526_arcsin__sin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X ) )
          = X ) ) ) ).

% arcsin_sin
thf(fact_8527_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_8528_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_8529_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_8530_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
              = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_8531_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_8532_cos__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_8533_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_8534_sin__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_8535_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_8536_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_8537_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X )
          = ( arccos @ Y ) )
        = ( X = Y ) ) ) ).

% arccos_eq_iff
thf(fact_8538_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% arccos_le_mono
thf(fact_8539_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_8540_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_8541_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% arccos_less_mono
thf(fact_8542_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_8543_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_8544_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_8545_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_8546_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_8547_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_8548_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_8549_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_8550_arccos__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).

% arccos_minus
thf(fact_8551_arccos__def,axiom,
    ( arccos
    = ( ^ [Y4: real] :
          ( the_real
          @ ^ [X4: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X4 )
              & ( ord_less_eq_real @ X4 @ pi )
              & ( ( cos_real @ X4 )
                = Y4 ) ) ) ) ) ).

% arccos_def
thf(fact_8552_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_8553_arccos__minus__abs,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X ) )
        = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).

% arccos_minus_abs
thf(fact_8554_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_8555_cot__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_8556_power__half__series,axiom,
    ( sums_real
    @ ^ [N6: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N6 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_8557_cot__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).

% cot_gt_zero
thf(fact_8558_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_8559_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_8560_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_8561_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_8562_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_8563_sinh__less__cosh__real,axiom,
    ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_less_cosh_real
thf(fact_8564_sinh__le__cosh__real,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_le_cosh_real
thf(fact_8565_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_8566_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_8567_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_8568_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_8569_cosh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( cosh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_8570_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_8571_eval__nat__numeral_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_8572_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq_int @ D5 @ Z2 )
                & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_8573_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq_int @ D5 @ Z8 )
                & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_8574_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_8575_Suc__0__mod__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat
        @ ( N
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_8576_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_8577_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
        = A4 ) ) ).

% set_encode_inverse
thf(fact_8578_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R ) )
      = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_8579_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_8580_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_8581_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q8: int,R5: int] : ( plus_plus_int @ Q8 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_8582_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% subset_decode_imp_le
thf(fact_8583_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_8584_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_8585_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = 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 @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( 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 @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_8586_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = 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 @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( 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 @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_8587_False__map2__or,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ $false @ bot_bot_set_o ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_o @ Xs ) )
       => ( ( map_Pr7541730621154948341_o_o_o @ ( produc6197397395684419436_o_o_o @ (|) ) @ ( zip_o_o @ Xs @ Ys ) )
          = Ys ) ) ) ).

% False_map2_or
thf(fact_8588_False__map2__and,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ $false @ bot_bot_set_o ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_o @ Xs ) )
       => ( ( map_Pr7541730621154948341_o_o_o @ ( produc6197397395684419436_o_o_o @ (&) ) @ ( zip_o_o @ Xs @ Ys ) )
          = Xs ) ) ) ).

% False_map2_and
thf(fact_8589_take__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_Suc_0
thf(fact_8590_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_8591_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_8592_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_8593_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_8594_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_8595_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_8596_not__take__bit__negative,axiom,
    ! [N: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_8597_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_8598_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% take_bit_nonnegative
thf(fact_8599_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_8600_take__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] :
      ( ( bit_se1745604003318907178nteger @ Xa2 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se2923211474154528505it_int @ Xa2 @ X ) ) ) ).

% take_bit_integer.abs_eq
thf(fact_8601_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_8602_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_8603_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_8604_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_8605_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_8606_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_8607_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_8608_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_8609_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_8610_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_8611_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_8612_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_8613_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N6: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N6 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N6 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N6 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_8614_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_8615_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_8616_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_8617_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_8618_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_8619_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_8620_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( last_nat @ ( upt @ I @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_8621_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_8622_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_8623_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).

% sum_list_upt
thf(fact_8624_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_8625_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append_nat @ Xs @ Ys ) )
        = ( ? [K3: nat] :
              ( ( ord_less_eq_nat @ I @ K3 )
              & ( ord_less_eq_nat @ K3 @ J )
              & ( ( upt @ I @ K3 )
                = Xs )
              & ( ( upt @ K3 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_8626_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_8627_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_8628_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_8629_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_8630_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_8631_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_8632_butlast__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( butlast_nat @ ( upt @ M @ N ) )
      = ( upt @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% butlast_upt
thf(fact_8633_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q3 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_8634_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_8635_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U3 )
     => ( ! [I2: nat] :
            ( ( ( ord_less_eq_nat @ U @ I2 )
              & ( ord_less_nat @ I2 @ U3 ) )
           => ~ ( P2 @ I2 ) )
       => ( ( filter_nat2 @ P2 @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat2 @ P2 @ ( upt @ zero_zero_nat @ U3 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_8636_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N6: nat] : ( minus_minus_nat @ N6 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_8637_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_8638_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N6: nat,M5: nat] : ( set_nat2 @ ( upt @ N6 @ ( suc @ M5 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_8639_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_8640_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N6: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N6 ) ) ) ) ).

% atLeast_upt
thf(fact_8641_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_8642_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N6: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N6 ) ) ) ) ) ).

% atMost_upto
thf(fact_8643_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_8644_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_8645_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_8646_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_8647_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N6: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N6 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N6 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N6 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_8648_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_8649_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_8650_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_8651_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_8652_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W2: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) @ N )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit0
thf(fact_8653_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W2: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) @ N )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit1
thf(fact_8654_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2923211474154528505it_int @ N @ L ) )
     => ( ( map_nat_o @ ( bit_se1146084159140164899it_int @ K ) @ ( upt @ zero_zero_nat @ N ) )
        = ( map_nat_o @ ( bit_se1146084159140164899it_int @ L ) @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% map_bit_range_eq_if_take_bit_eq
thf(fact_8655_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_8656_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N2 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).

% int_bit_bound
thf(fact_8657_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_8658_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_8659_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_8660_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_8661_bit__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bit_se9216721137139052372nteger @ ( code_integer_of_int @ X ) )
      = ( bit_se1146084159140164899it_int @ X ) ) ).

% bit_integer.abs_eq
thf(fact_8662_not__bit__Suc__0__numeral,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).

% not_bit_Suc_0_numeral
thf(fact_8663_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_nat_iff
thf(fact_8664_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_8665_num_Osize__gen_I2_J,axiom,
    ! [X2: num] :
      ( ( size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_8666_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
thf(fact_8667_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ 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 @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_8668_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_8669_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_8670_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_8671_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_8672_xor__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_8673_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_8674_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_8675_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_8676_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_8677_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N6: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N6 @ ( if_nat @ ( N6 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_8678_Suc__0__xor__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_8679_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N6: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N6 @ ( if_nat @ ( N6 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_8680_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_8681_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_8682_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_8683_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_8684_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_8685_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_8686_xor__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( bit_se3222712562003087583nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se6526347334894502574or_int @ Xa2 @ X ) ) ) ).

% xor_integer.abs_eq
thf(fact_8687_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_8688_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_8689_or__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% or_Suc_0_eq
thf(fact_8690_Suc__0__or__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_or_eq
thf(fact_8691_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_8692_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_8693_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_8694_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_8695_or__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( bit_se1080825931792720795nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se1409905431419307370or_int @ Xa2 @ X ) ) ) ).

% or_integer.abs_eq
thf(fact_8696_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_8697_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_8698_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( plus_plus_real @ X @ H6 ) @ S3 )
                 => ( ( ord_less_real @ H6 @ D2 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_8699_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( plus_plus_real @ X @ H6 ) @ S3 )
                 => ( ( ord_less_real @ H6 @ D2 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H6 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_8700_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F6: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F6 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z3: real] :
            ( ( ord_less_real @ A @ Z3 )
            & ( ord_less_real @ Z3 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F6 @ Z3 ) ) ) ) ) ) ).

% MVT2
thf(fact_8701_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D2 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H6 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_8702_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D2 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_8703_DERIV__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_8704_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_8705_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_8706_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G3: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G3 @ 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 @ ( G3 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_8707_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_8708_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_8709_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D2 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_8710_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D2 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H6 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_8711_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( minus_minus_real @ X @ H6 ) @ S3 )
                 => ( ( ord_less_real @ H6 @ D2 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H6 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_8712_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( minus_minus_real @ X @ H6 ) @ S3 )
                 => ( ( ord_less_real @ H6 @ D2 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_8713_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ( F @ X )
                = ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_8714_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_8715_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_8716_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_8717_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X4: real] : ( power_power_real @ X4 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_8718_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z2: real] : ( powr_real @ Z2 @ R )
        @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_8719_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ R )
          @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_8720_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_8721_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: real > real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_8722_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_8723_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_8724_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A4: set_real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A4 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_8725_artanh__real__has__field__derivative,axiom,
    ! [X: real,A4: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A4 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_8726_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_8727_DERIV__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_8728_DERIV__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_8729_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_8730_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_8731_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_8732_Maclaurin,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_8733_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_8734_Maclaurin__minus,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N )
                  & ( ord_less_eq_real @ H2 @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H2 @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_8735_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
                & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_8736_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,T6: real] :
            ( ( ( ord_less_nat @ M3 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_8737_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C2 )
           => ( ( ord_less_eq_real @ C2 @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C2 )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X @ C2 )
                         => ( ( ord_less_real @ X @ T6 )
                            & ( ord_less_real @ T6 @ C2 ) ) )
                        & ( ~ ( ord_less_real @ X @ C2 )
                         => ( ( ord_less_real @ C2 @ T6 )
                            & ( ord_less_real @ T6 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ M5 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_8738_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C2 )
           => ( ( ord_less_real @ C2 @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ C2 @ T6 )
                  & ( ord_less_real @ T6 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_8739_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C2 )
           => ( ( ord_less_eq_real @ C2 @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C2 )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_8740_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B4: real] :
      ( ! [M3: nat,T6: real] :
          ( ( ( ord_less_nat @ M3 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M2: nat,T7: real] :
            ( ( ( ord_less_nat @ M2 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P7: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P7 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P7 ) ) @ ( power_power_real @ U2 @ P7 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
                    @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P7: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P7 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P7 ) ) @ ( power_power_real @ T7 @ P7 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_8741_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X10: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_8742_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_8743_DERIV__power__series_H,axiom,
    ! [R2: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
         => ( summable_real
            @ ^ [N6: nat] : ( times_times_real @ ( times_times_real @ ( F @ N6 ) @ ( semiri5074537144036343181t_real @ ( suc @ N6 ) ) ) @ ( power_power_real @ X3 @ N6 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
       => ( ( ord_less_real @ zero_zero_real @ R2 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] :
                ( suminf_real
                @ ^ [N6: nat] : ( times_times_real @ ( F @ N6 ) @ ( power_power_real @ X4 @ ( suc @ N6 ) ) ) )
            @ ( suminf_real
              @ ^ [N6: nat] : ( times_times_real @ ( times_times_real @ ( F @ N6 ) @ ( semiri5074537144036343181t_real @ ( suc @ N6 ) ) ) @ ( power_power_real @ X0 @ N6 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_8744_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
           => ( ( F @ X )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_8745_DERIV__series_H,axiom,
    ! [F: real > nat > real,F6: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N2: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X4: real] : ( F @ X4 @ N2 )
          @ ( F6 @ X0 @ N2 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F6 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N2: nat,X3: real,Y3: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N2 ) @ ( F @ Y3 @ N2 ) ) ) @ ( times_times_real @ ( L5 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X4: real] : ( suminf_real @ ( F @ X4 ) )
                  @ ( suminf_real @ ( F6 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_8746_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L4: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L4 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L4 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_8747_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( code_int_of_integer @ ( ord_max_Code_integer @ K @ L ) )
      = ( ord_max_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).

% int_of_integer_max
thf(fact_8748_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_8749_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).

% finite_greaterThanLessThan_integer
thf(fact_8750_int__of__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% int_of_integer_of_nat
thf(fact_8751_integer__of__int__int__of__integer,axiom,
    ! [K: code_integer] :
      ( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
      = K ) ).

% integer_of_int_int_of_integer
thf(fact_8752_int__of__integer__integer__of__int,axiom,
    ! [K: int] :
      ( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
      = K ) ).

% int_of_integer_integer_of_int
thf(fact_8753_int__of__integer__inverse,axiom,
    ! [X: code_integer] :
      ( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
      = X ) ).

% int_of_integer_inverse
thf(fact_8754_of__int__integer__of,axiom,
    ! [K: code_integer] :
      ( ( ring_18347121197199848620nteger @ ( code_int_of_integer @ K ) )
      = K ) ).

% of_int_integer_of
thf(fact_8755_int__of__integer__of__int,axiom,
    ! [K: int] :
      ( ( code_int_of_integer @ ( ring_18347121197199848620nteger @ K ) )
      = K ) ).

% int_of_integer_of_int
thf(fact_8756_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_8757_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).

% card_greaterThanLessThan
thf(fact_8758_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_8759_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_8760_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( uminus1351360451143612070nteger @ X ) )
      = ( uminus_uminus_int @ ( code_int_of_integer @ X ) ) ) ).

% uminus_integer.rep_eq
thf(fact_8761_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa2 ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% plus_integer.rep_eq
thf(fact_8762_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa2 ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% minus_integer.rep_eq
thf(fact_8763_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% times_integer.rep_eq
thf(fact_8764_abs__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( abs_abs_Code_integer @ X ) )
      = ( abs_abs_int @ ( code_int_of_integer @ X ) ) ) ).

% abs_integer.rep_eq
thf(fact_8765_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa2 ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% divide_integer.rep_eq
thf(fact_8766_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X @ Xa2 ) )
      = ( modulo_modulo_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% modulo_integer.rep_eq
thf(fact_8767_sgn__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( sgn_sgn_Code_integer @ X ) )
      = ( sgn_sgn_int @ ( code_int_of_integer @ X ) ) ) ).

% sgn_integer.rep_eq
thf(fact_8768_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_8769_or__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1080825931792720795nteger @ X @ Xa2 ) )
      = ( bit_se1409905431419307370or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% or_integer.rep_eq
thf(fact_8770_bit__integer_Orep__eq,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [X4: code_integer] : ( bit_se1146084159140164899it_int @ ( code_int_of_integer @ X4 ) ) ) ) ).

% bit_integer.rep_eq
thf(fact_8771_int__of__integer__induct,axiom,
    ! [Y: int,P2: int > $o] :
      ( ( member_int @ Y @ top_top_set_int )
     => ( ! [X3: code_integer] : ( P2 @ ( code_int_of_integer @ X3 ) )
       => ( P2 @ Y ) ) ) ).

% int_of_integer_induct
thf(fact_8772_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( ( member_int @ Y @ top_top_set_int )
     => ~ ! [X3: code_integer] :
            ( Y
           != ( code_int_of_integer @ X3 ) ) ) ).

% int_of_integer_cases
thf(fact_8773_int__of__integer,axiom,
    ! [X: code_integer] : ( member_int @ ( code_int_of_integer @ X ) @ top_top_set_int ) ).

% int_of_integer
thf(fact_8774_type__definition__integer,axiom,
    type_d8366093980585677751er_int @ code_int_of_integer @ code_integer_of_int @ top_top_set_int ).

% type_definition_integer
thf(fact_8775_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( ( member_int @ Y @ top_top_set_int )
     => ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
        = Y ) ) ).

% integer_of_int_inverse
thf(fact_8776_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
      = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% int_of_integer_less_iff
thf(fact_8777_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_8778_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L4 ) ) ) ) ).

% integer_less_iff
thf(fact_8779_int__of__integer__inject,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( code_int_of_integer @ X )
        = ( code_int_of_integer @ Y ) )
      = ( X = Y ) ) ).

% int_of_integer_inject
thf(fact_8780_integer__eqI,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( code_int_of_integer @ K )
        = ( code_int_of_integer @ L ) )
     => ( K = L ) ) ).

% integer_eqI
thf(fact_8781_integer__eq__iff,axiom,
    ( ( ^ [Y6: code_integer,Z4: code_integer] : ( Y6 = Z4 ) )
    = ( ^ [K3: code_integer,L4: code_integer] :
          ( ( code_int_of_integer @ K3 )
          = ( code_int_of_integer @ L4 ) ) ) ) ).

% integer_eq_iff
thf(fact_8782_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_8783_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K3: code_integer,L4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L4 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_8784_xor__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3222712562003087583nteger @ X @ Xa2 ) )
      = ( bit_se6526347334894502574or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% xor_integer.rep_eq
thf(fact_8785_take__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1745604003318907178nteger @ X @ Xa2 ) )
      = ( bit_se2923211474154528505it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).

% take_bit_integer.rep_eq
thf(fact_8786_unset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se8260200283734997820nteger @ X @ Xa2 ) )
      = ( bit_se4203085406695923979it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).

% unset_bit_integer.rep_eq
thf(fact_8787_LIM__fun__less__zero,axiom,
    ! [F: real > real,L: real,C2: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X5 ) ) @ R4 ) )
               => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_8788_LIM__fun__not__zero,axiom,
    ! [F: real > real,L: real,C2: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
     => ( ( L != zero_zero_real )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X5 ) ) @ R4 ) )
               => ( ( F @ X5 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_8789_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L: real,C2: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X5 ) ) @ R4 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_8790_Code__Numeral_Oset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se2793503036327961859nteger @ X @ Xa2 ) )
      = ( bit_se7879613467334960850it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).

% Code_Numeral.set_bit_integer.rep_eq
thf(fact_8791_flip__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1345352211410354436nteger @ X @ Xa2 ) )
      = ( bit_se2159334234014336723it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).

% flip_bit_integer.rep_eq
thf(fact_8792_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N6: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N6 ) @ M5 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_8793_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or4266950643985792945nteger @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_8794_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_8795_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_8796_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L4: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L4 ) @ ( modulo364778990260209775nteger @ K3 @ L4 ) ) ) ) ).

% divmod_integer_def
thf(fact_8797_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_8798_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_8799_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_8800_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( filterlim_nat_nat @ ( times_times_nat @ C2 ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_8801_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( filterlim_nat_nat
        @ ^ [X4: nat] : ( times_times_nat @ X4 @ C2 )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_8802_monoseq__convergent,axiom,
    ! [X6: nat > real,B4: real] :
      ( ( topolo6980174941875973593q_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X6 @ I2 ) ) @ B4 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_8803_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
         => ( ( filterlim_nat_real
              @ ^ [N6: nat] : ( minus_minus_real @ ( F @ N6 ) @ ( G @ N6 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L2: real] :
                ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L2 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
                & ! [N9: nat] : ( ord_less_eq_real @ L2 @ ( G @ N9 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_8804_LIMSEQ__inverse__zero,axiom,
    ! [X6: nat > real] :
      ( ! [R4: real] :
        ? [N10: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N10 @ N2 )
         => ( ord_less_real @ R4 @ ( X6 @ N2 ) ) )
     => ( filterlim_nat_real
        @ ^ [N6: nat] : ( inverse_inverse_real @ ( X6 @ N6 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_8805_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N6: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N6 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_8806_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( filterlim_nat_real
        @ ^ [N6: nat] : ( root @ N6 @ C2 )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_8807_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N6: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N6 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_8808_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_8809_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_8810_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N6: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N6 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_8811_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C2 ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_8812_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_8813_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N6: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N6 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_8814_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N6: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N6 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_8815_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N6: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N6 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_8816_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( summable_real
            @ ^ [N6: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N6 ) @ ( A @ N6 ) ) ) ) ) ) ).

% summable
thf(fact_8817_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N6: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_8818_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_8819_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N6: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_8820_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L2: real] :
              ( ! [N9: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                  @ L2 )
              & ( filterlim_nat_real
                @ ^ [N6: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                @ ( topolo2815343760600316023s_real @ L2 )
                @ at_top_nat )
              & ! [N9: nat] :
                  ( ord_less_eq_real @ L2
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N6: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L2 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_8821_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_8822_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N6: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_8823_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L4: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L4 ) @ ( code_num_of_integer @ L4 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L4 ) @ ( code_num_of_integer @ L4 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_8824_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L4: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L4 ) @ ( code_nat_of_integer @ L4 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L4 ) @ ( code_nat_of_integer @ L4 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_8825_nat__of__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_nat_of_integer @ ( semiri4939895301339042750nteger @ N ) )
      = N ) ).

% nat_of_integer_of_nat
thf(fact_8826_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_8827_of__nat__of__integer,axiom,
    ! [K: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% of_nat_of_integer
thf(fact_8828_nat__of__integer_Orep__eq,axiom,
    ( code_nat_of_integer
    = ( ^ [X4: code_integer] : ( nat2 @ ( code_int_of_integer @ X4 ) ) ) ) ).

% nat_of_integer.rep_eq
thf(fact_8829_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
      = ( nat2 @ X ) ) ).

% nat_of_integer.abs_eq
thf(fact_8830_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_8831_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
          = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).

% nat_of_integer_less_iff
thf(fact_8832_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_8833_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
            @ at_bot_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_8834_eventually__sequentially__Suc,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat
        @ ^ [I3: nat] : ( P2 @ ( suc @ I3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_8835_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).

% at_bot_le_at_infinity
thf(fact_8836_eventually__sequentiallyI,axiom,
    ! [C2: nat,P2: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C2 @ X3 )
         => ( P2 @ X3 ) )
     => ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_8837_eventually__sequentially,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat @ P2 @ at_top_nat )
      = ( ? [N7: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N7 @ N6 )
           => ( P2 @ N6 ) ) ) ) ).

% eventually_sequentially
thf(fact_8838_le__sequentially,axiom,
    ! [F2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
      = ( ! [N7: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N7 ) @ F2 ) ) ) ).

% le_sequentially
thf(fact_8839_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( eventually_real
        @ ^ [X4: real] : ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_8840_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ X3 @ B )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_8841_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).

% at_top_le_at_infinity
thf(fact_8842_Bseq__eq__bounded,axiom,
    ! [F: nat > real,A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
     => ( bfun_nat_real @ F @ at_top_nat ) ) ).

% Bseq_eq_bounded
thf(fact_8843_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_8844_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B @ X3 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_8845_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
            @ at_top_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_8846_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_8847_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_8848_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N6: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N6 ) @ ( suc @ M5 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_8849_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_8850_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_8851_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ 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
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_8852_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ 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
                    @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X4: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                              & ! [X4: nat] :
                                  ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                   => ( ( ord_less_nat @ Mi3 @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima2 ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_8853_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).

% finite_greaterThanAtMost_integer
thf(fact_8854_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or2715278749043346189nteger @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_8855_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima3: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima3 @ Deg @ TreeList @ Summary ) @ Deg3 )
      = ( ( Deg = Deg3 )
        & ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ 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
          @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
            & ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                    & ! [X4: nat] :
                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima3 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_8856_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [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
                  @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                    & ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I3: nat] :
                            ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                            & ! [X4: nat] :
                                ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                 => ( ( ord_less_nat @ Mi3 @ X4 )
                                    & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima2 ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_8857_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [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
                      @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                        & ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I3: nat] :
                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                & ! [X4: nat] :
                                    ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                     => ( ( ord_less_nat @ Mi3 @ X4 )
                                        & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_8858_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ 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
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_8859_Sup__real__def,axiom,
    ( comple1385675409528146559p_real
    = ( ^ [X8: set_real] :
          ( ord_Least_real
          @ ^ [Z2: real] :
            ! [X4: real] :
              ( ( member_real @ X4 @ X8 )
             => ( ord_less_eq_real @ X4 @ Z2 ) ) ) ) ) ).

% Sup_real_def
thf(fact_8860_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X8: set_int] :
          ( the_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ X8 )
              & ! [Y4: int] :
                  ( ( member_int @ Y4 @ X8 )
                 => ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_8861_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
         => ~ ! [Mima2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ 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
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima2 ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_8862_set__bits__int__def,axiom,
    ( bit_bi6516823479961619367ts_int
    = ( ^ [F5: nat > $o] :
          ( if_int
          @ ? [N6: nat] :
            ! [M5: nat] :
              ( ( ord_less_eq_nat @ N6 @ M5 )
             => ( ( F5 @ M5 )
                = ( F5 @ N6 ) ) )
          @ ( bit_ri631733984087533419it_int
            @ ( ord_Least_nat
              @ ^ [N6: nat] :
                ! [M5: nat] :
                  ( ( ord_less_eq_nat @ N6 @ M5 )
                 => ( ( F5 @ M5 )
                    = ( F5 @ N6 ) ) ) )
            @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
              @ ( map_nat_o @ F5
                @ ( upt @ zero_zero_nat
                  @ ( suc
                    @ ( ord_Least_nat
                      @ ^ [N6: nat] :
                        ! [M5: nat] :
                          ( ( ord_less_eq_nat @ N6 @ M5 )
                         => ( ( F5 @ M5 )
                            = ( F5 @ N6 ) ) ) ) ) ) ) ) )
          @ zero_zero_int ) ) ) ).

% set_bits_int_def
thf(fact_8863_set__bits__int__unfold_H,axiom,
    ( bit_bi6516823479961619367ts_int
    = ( ^ [F5: nat > $o] :
          ( if_int
          @ ? [N6: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N6 @ N11 )
             => ~ ( F5 @ N11 ) )
          @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
            @ ( map_nat_o @ F5
              @ ( upt @ zero_zero_nat
                @ ( ord_Least_nat
                  @ ^ [N6: nat] :
                    ! [N11: nat] :
                      ( ( ord_less_eq_nat @ N6 @ N11 )
                     => ~ ( F5 @ N11 ) ) ) ) ) )
          @ ( if_int
            @ ? [N6: nat] :
              ! [N11: nat] :
                ( ( ord_less_eq_nat @ N6 @ N11 )
               => ( F5 @ N11 ) )
            @ ( bit_ri631733984087533419it_int
              @ ( ord_Least_nat
                @ ^ [N6: nat] :
                  ! [N11: nat] :
                    ( ( ord_less_eq_nat @ N6 @ N11 )
                   => ( F5 @ N11 ) ) )
              @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
                @ ( append_o
                  @ ( map_nat_o @ F5
                    @ ( upt @ zero_zero_nat
                      @ ( ord_Least_nat
                        @ ^ [N6: nat] :
                          ! [N11: nat] :
                            ( ( ord_less_eq_nat @ N6 @ N11 )
                           => ( F5 @ N11 ) ) ) ) )
                  @ ( cons_o @ $true @ nil_o ) ) ) )
            @ zero_zero_int ) ) ) ) ).

% set_bits_int_unfold'
thf(fact_8864_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_8865_wf__set__bits__int__Suc,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int
        @ ^ [N6: nat] : ( F @ ( suc @ N6 ) ) )
      = ( bit_wf_set_bits_int @ F ) ) ).

% wf_set_bits_int_Suc
thf(fact_8866_wf__set__bits__int__simps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F5: nat > $o] :
        ? [N6: nat] :
          ( ! [N11: nat] :
              ( ( ord_less_eq_nat @ N6 @ N11 )
             => ~ ( F5 @ N11 ) )
          | ! [N11: nat] :
              ( ( ord_less_eq_nat @ N6 @ N11 )
             => ( F5 @ N11 ) ) ) ) ) ).

% wf_set_bits_int_simps
thf(fact_8867_zeros,axiom,
    ! [N: nat,F: nat > $o] :
      ( ! [N5: nat] :
          ( ( ord_less_eq_nat @ N @ N5 )
         => ~ ( F @ N5 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% zeros
thf(fact_8868_wf__set__bits__int_Osimps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F5: nat > $o] :
          ( ? [N6: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N6 @ N11 )
             => ~ ( F5 @ N11 ) )
          | ? [N6: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N6 @ N11 )
             => ( F5 @ N11 ) ) ) ) ) ).

% wf_set_bits_int.simps
thf(fact_8869_wf__set__bits__int_Ocases,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ! [N2: nat] :
            ~ ! [N4: nat] :
                ( ( ord_less_eq_nat @ N2 @ N4 )
               => ~ ( F @ N4 ) )
       => ~ ! [N2: nat] :
              ~ ! [N4: nat] :
                  ( ( ord_less_eq_nat @ N2 @ N4 )
                 => ( F @ N4 ) ) ) ) ).

% wf_set_bits_int.cases
thf(fact_8870_ones,axiom,
    ! [N: nat,F: nat > $o] :
      ( ! [N5: nat] :
          ( ( ord_less_eq_nat @ N @ N5 )
         => ( F @ N5 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% ones
thf(fact_8871_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_8872_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_8873_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
          @ ^ [B7: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B7 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B7: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B7 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C62 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_eq_char_simp
thf(fact_8874_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
          @ ^ [B7: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B7 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B7: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B7 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C62 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_char_simp
thf(fact_8875_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_8876_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S3 ) ) ) ) ).

% infinite_int_iff_infinite_nat_abs
thf(fact_8877_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).

% Code_Numeral.negative_def
thf(fact_8878_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L4: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L4
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L4 ) )
                @ ( code_divmod_abs @ K3 @ L4 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L4 ) @ S8 ) ) )
                  @ ( code_divmod_abs @ K3 @ L4 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_8879_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_8880_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_8881_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L4: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L4 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L4 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L4 @ S8 ) ) )
                @ ( code_divmod_abs @ K3 @ L4 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L4 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L4 ) @ S8 ) ) )
                    @ ( code_divmod_abs @ K3 @ L4 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_8882_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K3: code_integer,L4: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L4 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L4 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_8883_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S8: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S8 ) ) @ ( S8 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_8884_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_8885_len__num0,axiom,
    ( type_l4264026598287037464l_num0
    = ( ^ [Uu4: itself_Numeral_num0] : zero_zero_nat ) ) ).

% len_num0
thf(fact_8886_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_8887_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_8888_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_8889_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_8890_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_8891_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_8892_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_8893_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_8894_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_8895_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_8896_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_8897_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_8898_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_8899_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_8900_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_8901_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_8902_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_8903_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).

% min_diff
thf(fact_8904_int__of__integer__min,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( code_int_of_integer @ ( ord_min_Code_integer @ K @ L ) )
      = ( ord_min_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).

% int_of_integer_min
thf(fact_8905_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_8906_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_8907_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_8908_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_8909_drop__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_8910_drop__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] :
      ( ( bit_se3928097537394005634nteger @ Xa2 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se8568078237143864401it_int @ Xa2 @ X ) ) ) ).

% drop_bit_integer.abs_eq
thf(fact_8911_drop__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3928097537394005634nteger @ X @ Xa2 ) )
      = ( bit_se8568078237143864401it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).

% drop_bit_integer.rep_eq
thf(fact_8912_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% mask_nat_positive_iff
thf(fact_8913_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_8914_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_8915_push__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_Suc_0
thf(fact_8916_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_8917_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_8918_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_8919_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_8920_mask__integer_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_int_of_integer @ ( bit_se2119862282449309892nteger @ X ) )
      = ( bit_se2000444600071755411sk_int @ X ) ) ).

% mask_integer.rep_eq
thf(fact_8921_push__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se7788150548672797655nteger @ X @ Xa2 ) )
      = ( bit_se545348938243370406it_int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).

% push_bit_integer.rep_eq
thf(fact_8922_mask__integer_Oabs__eq,axiom,
    ( bit_se2119862282449309892nteger
    = ( ^ [X4: nat] : ( code_integer_of_int @ ( bit_se2000444600071755411sk_int @ X4 ) ) ) ) ).

% mask_integer.abs_eq
thf(fact_8923_push__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] :
      ( ( bit_se7788150548672797655nteger @ Xa2 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se545348938243370406it_int @ Xa2 @ X ) ) ) ).

% push_bit_integer.abs_eq
thf(fact_8924_less__mask,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% less_mask
thf(fact_8925_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_8926_Suc__mask__eq__exp,axiom,
    ! [N: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_mask_eq_exp
thf(fact_8927_mask__nat__less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% mask_nat_less_exp
thf(fact_8928_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_8929_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_8930_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_8931_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_8932_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_8933_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_8934_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_8935_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_8936_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_8937_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_8938_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_int_code(2)
thf(fact_8939_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_8940_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_8941_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_8942_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_8943_and__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_8944_Suc__0__and__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_8945_and__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3949692690581998587nteger @ X @ Xa2 ) )
      = ( bit_se725231765392027082nd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% and_integer.rep_eq
thf(fact_8946_and__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( bit_se3949692690581998587nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se725231765392027082nd_int @ Xa2 @ X ) ) ) ).

% and_integer.abs_eq
thf(fact_8947_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N6: nat] :
          ( if_nat
          @ ( ( M5 = zero_zero_nat )
            | ( N6 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_8948_drop__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se8568078237143864401it_int @ zero_zero_nat @ I )
      = I ) ).

% drop_bit_int_code(1)
thf(fact_8949_push__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
      = I ) ).

% push_bit_int_code(1)
thf(fact_8950_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_8951_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_8952_not__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bit_ri7632146776885996613nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_ri7919022796975470100ot_int @ X ) ) ) ).

% not_integer.abs_eq
thf(fact_8953_not__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( bit_ri7919022796975470100ot_int @ ( code_int_of_integer @ X ) ) ) ).

% not_integer.rep_eq
thf(fact_8954_shiftl__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% shiftl_Suc_0
thf(fact_8955_shiftr__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh2154871086232339855tr_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% shiftr_Suc_0
thf(fact_8956_not__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ bit_ri7919022796975470100ot_int
    @ bit_ri7919022796975470100ot_int ) ).

% not_integer.rsp
thf(fact_8957_xor__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se6526347334894502574or_int
    @ bit_se6526347334894502574or_int ) ).

% xor_integer.rsp
thf(fact_8958_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_8959_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_8960_minus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ minus_minus_int
    @ minus_minus_int ) ).

% minus_integer.rsp
thf(fact_8961_dup_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).

% dup.rsp
thf(fact_8962_bit__integer_Orsp,axiom,
    ( bNF_re3376528473927230327_nat_o
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [Y6: nat > $o,Z4: nat > $o] : ( Y6 = Z4 )
    @ bit_se1146084159140164899it_int
    @ bit_se1146084159140164899it_int ) ).

% bit_integer.rsp
thf(fact_8963_abs__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ abs_abs_int
    @ abs_abs_int ) ).

% abs_integer.rsp
thf(fact_8964_sgn__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ sgn_sgn_int
    @ sgn_sgn_int ) ).

% sgn_integer.rsp
thf(fact_8965_natural__of__integer_Orsp,axiom,
    ( bNF_re3715656647883201625at_nat
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ nat2
    @ nat2 ) ).

% natural_of_integer.rsp
thf(fact_8966_divide__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ divide_divide_int
    @ divide_divide_int ) ).

% divide_integer.rsp
thf(fact_8967_modulo__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ modulo_modulo_int
    @ modulo_modulo_int ) ).

% modulo_integer.rsp
thf(fact_8968_uminus__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ uminus_uminus_int
    @ uminus_uminus_int ) ).

% uminus_integer.rsp
thf(fact_8969_plus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ plus_plus_int
    @ plus_plus_int ) ).

% plus_integer.rsp
thf(fact_8970_times__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ times_times_int
    @ times_times_int ) ).

% times_integer.rsp
thf(fact_8971_or__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se1409905431419307370or_int
    @ bit_se1409905431419307370or_int ) ).

% or_integer.rsp
thf(fact_8972_and__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se725231765392027082nd_int
    @ bit_se725231765392027082nd_int ) ).

% and_integer.rsp
thf(fact_8973_xor__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se6528837805403552850or_nat
    @ bit_se6528837805403552850or_nat ) ).

% xor_natural.rsp
thf(fact_8974_drop__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se8568078237143864401it_int
    @ bit_se8568078237143864401it_int ) ).

% drop_bit_integer.rsp
thf(fact_8975_drop__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se8570568707652914677it_nat
    @ bit_se8570568707652914677it_nat ) ).

% drop_bit_natural.rsp
thf(fact_8976_take__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se2923211474154528505it_int
    @ bit_se2923211474154528505it_int ) ).

% take_bit_integer.rsp
thf(fact_8977_take__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se2925701944663578781it_nat
    @ bit_se2925701944663578781it_nat ) ).

% take_bit_natural.rsp
thf(fact_8978_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_8979_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_8980_inj__on__diff__nat,axiom,
    ! [N8: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N8 )
         => ( ord_less_eq_nat @ K @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N6: nat] : ( minus_minus_nat @ N6 @ K )
        @ N8 ) ) ).

% inj_on_diff_nat
thf(fact_8981_minus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ minus_minus_nat
    @ minus_minus_nat ) ).

% minus_natural.rsp
thf(fact_8982_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ ^ [M5: num,N6: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N6 ) )
    @ ^ [M5: num,N6: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N6 ) ) ) ).

% sub.rsp
thf(fact_8983_bit__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ^ [Y6: nat > $o,Z4: nat > $o] : ( Y6 = Z4 )
    @ bit_se1148574629649215175it_nat
    @ bit_se1148574629649215175it_nat ) ).

% bit_natural.rsp
thf(fact_8984_unset__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se4203085406695923979it_int
    @ bit_se4203085406695923979it_int ) ).

% unset_bit_integer.rsp
thf(fact_8985_flip__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se2159334234014336723it_int
    @ bit_se2159334234014336723it_int ) ).

% flip_bit_integer.rsp
thf(fact_8986_inj__Suc,axiom,
    ! [N8: set_nat] : ( inj_on_nat_nat @ suc @ N8 ) ).

% inj_Suc
thf(fact_8987_Suc_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ suc
    @ suc ) ).

% Suc.rsp
thf(fact_8988_divide__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ divide_divide_nat
    @ divide_divide_nat ) ).

% divide_natural.rsp
thf(fact_8989_modulo__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ modulo_modulo_nat
    @ modulo_modulo_nat ) ).

% modulo_natural.rsp
thf(fact_8990_Code__Numeral_Oset__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se7879613467334960850it_int
    @ bit_se7879613467334960850it_int ) ).

% Code_Numeral.set_bit_integer.rsp
thf(fact_8991_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_8992_times__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ times_times_nat
    @ times_times_nat ) ).

% times_natural.rsp
thf(fact_8993_or__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se1412395901928357646or_nat
    @ bit_se1412395901928357646or_nat ) ).

% or_natural.rsp
thf(fact_8994_integer__of__natural_Orsp,axiom,
    ( bNF_re6650684261131312217nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ semiri1314217659103216013at_int
    @ semiri1314217659103216013at_int ) ).

% integer_of_natural.rsp
thf(fact_8995_inj__on__set__encode,axiom,
    inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).

% inj_on_set_encode
thf(fact_8996_mask__integer_Orsp,axiom,
    ( bNF_re6650684261131312217nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
    @ bit_se2000444600071755411sk_int
    @ bit_se2000444600071755411sk_int ) ).

% mask_integer.rsp
thf(fact_8997_mask__natural_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ bit_se2002935070580805687sk_nat
    @ bit_se2002935070580805687sk_nat ) ).

% mask_natural.rsp
thf(fact_8998_push__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
      @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    @ bit_se545348938243370406it_int
    @ bit_se545348938243370406it_int ) ).

% push_bit_integer.rsp
thf(fact_8999_push__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se547839408752420682it_nat
    @ bit_se547839408752420682it_nat ) ).

% push_bit_natural.rsp
thf(fact_9000_and__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se727722235901077358nd_nat
    @ bit_se727722235901077358nd_nat ) ).

% and_natural.rsp
thf(fact_9001_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_9002_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_9003_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_9004_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y4: real] : ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9005_unset__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se4205575877204974255it_nat
    @ bit_se4205575877204974255it_nat ) ).

% unset_bit_natural.rsp
thf(fact_9006_flip__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se2161824704523386999it_nat
    @ bit_se2161824704523386999it_nat ) ).

% flip_bit_natural.rsp
thf(fact_9007_set__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
      @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    @ bit_se7882103937844011126it_nat
    @ bit_se7882103937844011126it_nat ) ).

% set_bit_natural.rsp
thf(fact_9008_integer__of__num_I3_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_9009_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
thf(fact_9010_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9011_integer__of__num_I2_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).

% integer_of_num(2)
thf(fact_9012_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9013_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_9014_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_9015_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( divide_divide_nat @ M @ N ) ) ).

% fst_divmod_nat
thf(fact_9016_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
      = ( divide6298287555418463151nteger @ K @ L ) ) ).

% fst_divmod_integer
thf(fact_9017_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_9018_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9019_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X4: nat,Y4: nat] : ( if_Pro3027730157355071871nt_int @ ( Y4 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y4 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9020_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9021_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9022_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( modulo_modulo_nat @ M @ N ) ) ).

% snd_divmod_nat
thf(fact_9023_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L ) )
      = ( modulo364778990260209775nteger @ K @ L ) ) ).

% snd_divmod_integer
thf(fact_9024_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_9025_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_9026_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_9027_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H2 )
      = ( ( produc8586169260539613262et_nat @ H2 )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_9028_mod__star__trueE_H,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ 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 @ P2 @ H5 ) ) ) ) ).

% mod_star_trueE'
thf(fact_9029_one__mod__minus__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9030_minus__one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9031_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9032_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9033_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L4: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L4 @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_9034_normalize__def,axiom,
    ( normalize
    = ( ^ [P7: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P7 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P7 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P7 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P7 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P7 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P7 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9035_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
      = ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9036_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_9037_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_9038_gcd__cases__int,axiom,
    ! [X: int,Y: int,P2: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P2 @ ( gcd_gcd_int @ X @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P2 @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P2 @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P2 @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P2 @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9039_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_ge_0_int
thf(fact_9040_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_9041_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ Y )
     => ( ( gcd_gcd_int @ X @ Y )
        = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).

% gcd_non_0_int
thf(fact_9042_integer__set__bit__code,axiom,
    ( bits_integer_set_bit
    = ( ^ [X4: code_integer,N6: code_integer,B7: $o] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger ) @ ( undefi1878487536576149250nteger @ X4 @ N6 @ B7 ) @ ( if_Code_integer @ B7 @ ( bit_se1080825931792720795nteger @ X4 @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N6 ) @ one_one_Code_integer ) ) @ ( bit_se3949692690581998587nteger @ X4 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N6 ) @ one_one_Code_integer ) ) ) ) ) ) ) ).

% integer_set_bit_code
thf(fact_9043_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( gcd_gcd_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9044_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9045_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9046_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9047_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_9048_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_9049_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9050_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
      = ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9051_Gcd__in,axiom,
    ! [A4: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( member_nat @ A3 @ A4 )
         => ( ( member_nat @ B3 @ A4 )
           => ( member_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ A4 ) ) )
     => ( ( A4 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A4 ) @ A4 ) ) ) ).

% Gcd_in
thf(fact_9052_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_9053_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_9054_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_9055_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9056_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9057_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_9058_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9059_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X4: nat,Y4: nat] : ( if_nat @ ( Y4 = zero_zero_nat ) @ X4 @ ( gcd_gcd_nat @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9060_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X @ Y )
        = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9061_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M5: nat,N6: nat] :
        ( ( dvd_dvd_nat @ M5 @ N6 )
        & ( M5 != N6 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9062_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] :
                ( ( dvd_dvd_nat @ D5 @ M )
                & ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9063_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9064_uint32__shiftl__def,axiom,
    ( uint32_shiftl
    = ( ^ [X4: uint32,N6: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N6 ) )
          @ ( undefi8952517107220742160uint32 @ bit_se5742574853984576102uint32 @ X4 @ N6 )
          @ ( bit_se5742574853984576102uint32 @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).

% uint32_shiftl_def
thf(fact_9065_uint32__shiftr__def,axiom,
    ( uint32_shiftr
    = ( ^ [X4: uint32,N6: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N6 ) )
          @ ( undefi8952517107220742160uint32 @ bit_se3964402333458159761uint32 @ X4 @ N6 )
          @ ( bit_se3964402333458159761uint32 @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).

% uint32_shiftr_def
thf(fact_9066_uint32__test__bit__def,axiom,
    ( uint32_test_bit
    = ( ^ [X4: uint32,N6: code_integer] :
          ( ( ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
              | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N6 ) )
           => ( undefi6981832269580975664eger_o @ bit_se5367290876889521763uint32 @ X4 @ N6 ) )
          & ( ~ ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
                | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N6 ) )
           => ( bit_se5367290876889521763uint32 @ X4 @ ( code_nat_of_integer @ N6 ) ) ) ) ) ) ).

% uint32_test_bit_def
thf(fact_9067_integer__set__bit__def,axiom,
    ( bits_integer_set_bit
    = ( ^ [X4: code_integer,N6: code_integer,B7: $o] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger ) @ ( undefi1878487536576149250nteger @ X4 @ N6 @ B7 ) @ ( generi2397576812484419408nteger @ X4 @ ( code_nat_of_integer @ N6 ) @ B7 ) ) ) ) ).

% integer_set_bit_def
thf(fact_9068_uint32__shiftr__code,axiom,
    ! [N: code_integer,W2: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_shiftr @ W2 @ N ) )
          = ( rep_uint32 @ ( undefi8952517107220742160uint32 @ bit_se3964402333458159761uint32 @ W2 @ N ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_shiftr @ W2 @ N ) )
          = ( bit_se5176125413884933531l_num1 @ ( code_nat_of_integer @ N ) @ ( rep_uint32 @ W2 ) ) ) ) ) ).

% uint32_shiftr_code
thf(fact_9069_uint32_Oless__iff__word__of,axiom,
    ( ord_less_uint32
    = ( ^ [P7: uint32,Q8: uint32] : ( ord_le750835935415966154l_num1 @ ( rep_uint32 @ P7 ) @ ( rep_uint32 @ Q8 ) ) ) ) ).

% uint32.less_iff_word_of
thf(fact_9070_less__uint32_Orep__eq,axiom,
    ( ord_less_uint32
    = ( ^ [X4: uint32,Xa4: uint32] : ( ord_le750835935415966154l_num1 @ ( rep_uint32 @ X4 ) @ ( rep_uint32 @ Xa4 ) ) ) ) ).

% less_uint32.rep_eq
thf(fact_9071_uint32_Oless__eq__iff__word__of,axiom,
    ( ord_less_eq_uint32
    = ( ^ [P7: uint32,Q8: uint32] : ( ord_le3335648743751981014l_num1 @ ( rep_uint32 @ P7 ) @ ( rep_uint32 @ Q8 ) ) ) ) ).

% uint32.less_eq_iff_word_of
thf(fact_9072_less__eq__uint32_Orep__eq,axiom,
    ( ord_less_eq_uint32
    = ( ^ [X4: uint32,Xa4: uint32] : ( ord_le3335648743751981014l_num1 @ ( rep_uint32 @ X4 ) @ ( rep_uint32 @ Xa4 ) ) ) ) ).

% less_eq_uint32.rep_eq
thf(fact_9073_uint32__test__bit__code,axiom,
    ( uint32_test_bit
    = ( ^ [W3: uint32,N6: code_integer] :
          ( ( ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
              | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N6 ) )
           => ( undefi6981832269580975664eger_o @ bit_se5367290876889521763uint32 @ W3 @ N6 ) )
          & ( ~ ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
                | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N6 ) )
           => ( bit_se6859397288646540909l_num1 @ ( rep_uint32 @ W3 ) @ ( code_nat_of_integer @ N6 ) ) ) ) ) ) ).

% uint32_test_bit_code
thf(fact_9074_uint32__shiftl__code,axiom,
    ! [N: code_integer,W2: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_shiftl @ W2 @ N ) )
          = ( rep_uint32 @ ( undefi8952517107220742160uint32 @ bit_se5742574853984576102uint32 @ W2 @ N ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_shiftl @ W2 @ N ) )
          = ( bit_se837345729053750000l_num1 @ ( code_nat_of_integer @ N ) @ ( rep_uint32 @ W2 ) ) ) ) ) ).

% uint32_shiftl_code
thf(fact_9075_integer__shiftl__def,axiom,
    ( bits_integer_shiftl
    = ( ^ [X4: code_integer,N6: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger ) @ ( undefi8133104259855420269nteger @ X4 @ N6 ) @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).

% integer_shiftl_def
thf(fact_9076_less__eq__uint32_Orsp,axiom,
    ( bNF_re7364608769721783435num1_o
    @ ^ [Y6: word_N3645301735248828278l_num1,Z4: word_N3645301735248828278l_num1] : ( Y6 = Z4 )
    @ ( bNF_re5013357767504289739m1_o_o
      @ ^ [Y6: word_N3645301735248828278l_num1,Z4: word_N3645301735248828278l_num1] : ( Y6 = Z4 )
      @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
    @ ord_le3335648743751981014l_num1
    @ ord_le3335648743751981014l_num1 ) ).

% less_eq_uint32.rsp
thf(fact_9077_less__uint32_Orsp,axiom,
    ( bNF_re7364608769721783435num1_o
    @ ^ [Y6: word_N3645301735248828278l_num1,Z4: word_N3645301735248828278l_num1] : ( Y6 = Z4 )
    @ ( bNF_re5013357767504289739m1_o_o
      @ ^ [Y6: word_N3645301735248828278l_num1,Z4: word_N3645301735248828278l_num1] : ( Y6 = Z4 )
      @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
    @ ord_le750835935415966154l_num1
    @ ord_le750835935415966154l_num1 ) ).

% less_uint32.rsp
thf(fact_9078_uint32__set__bit__code,axiom,
    ! [N: code_integer,W2: uint32,B: $o] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_set_bit @ W2 @ N @ B ) )
          = ( rep_uint32 @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ W2 @ N @ B ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_set_bit @ W2 @ N @ B ) )
          = ( generi5268133209446125161l_num1 @ ( rep_uint32 @ W2 ) @ ( code_nat_of_integer @ N ) @ B ) ) ) ) ).

% uint32_set_bit_code
thf(fact_9079_uint32__set__bit__def,axiom,
    ( uint32_set_bit
    = ( ^ [X4: uint32,N6: code_integer,B7: $o] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N6 ) )
          @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ X4 @ N6 @ B7 )
          @ ( generi1993664874377053279uint32 @ X4 @ ( code_nat_of_integer @ N6 ) @ B7 ) ) ) ) ).

% uint32_set_bit_def
thf(fact_9080_integer__shiftr__def,axiom,
    ( bits_integer_shiftr
    = ( ^ [X4: code_integer,N6: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger ) @ ( undefi8133104259855420269nteger @ X4 @ N6 ) @ ( bit_se3928097537394005634nteger @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).

% integer_shiftr_def
thf(fact_9081_uint32__divmod__code,axiom,
    ( uint32_divmod
    = ( ^ [X4: uint32,Y4: 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y4 ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_uint32 @ X4 @ Y4 ) @ ( produc1400373151660368625uint32 @ zero_zero_uint32 @ X4 ) @ ( produc1400373151660368625uint32 @ one_one_uint32 @ ( minus_minus_uint32 @ X4 @ Y4 ) ) ) @ ( if_Pro1135515155860407935uint32 @ ( Y4 = zero_zero_uint32 ) @ ( produc1400373151660368625uint32 @ ( div0_uint32 @ X4 ) @ ( mod0_uint32 @ X4 ) ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_eq_uint32 @ Y4 @ ( minus_minus_uint32 @ X4 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X4 ) @ Y4 ) ) @ Y4 ) ) ) @ ( produc1400373151660368625uint32 @ ( plus_plus_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X4 ) @ Y4 ) ) @ one_one_uint32 ) @ ( minus_minus_uint32 @ ( minus_minus_uint32 @ X4 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X4 ) @ Y4 ) ) @ Y4 ) ) @ Y4 ) ) @ ( produc1400373151660368625uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X4 ) @ Y4 ) ) @ ( minus_minus_uint32 @ X4 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X4 ) @ Y4 ) ) @ Y4 ) ) ) ) ) ) ) ) ).

% uint32_divmod_code
thf(fact_9082_uint32_Oset__bits__aux__code,axiom,
    ( set_bits_aux_uint32
    = ( ^ [F5: nat > $o,N6: nat,W3: uint32] : ( if_uint32 @ ( N6 = zero_zero_nat ) @ W3 @ ( set_bits_aux_uint32 @ F5 @ ( minus_minus_nat @ N6 @ one_one_nat ) @ ( bit_se2966626333419230250uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ W3 ) @ ( if_uint32 @ ( F5 @ ( minus_minus_nat @ N6 @ one_one_nat ) ) @ one_one_uint32 @ zero_zero_uint32 ) ) ) ) ) ) ).

% uint32.set_bits_aux_code
thf(fact_9083_shiftr__uint32__code,axiom,
    ( bit_se3964402333458159761uint32
    = ( ^ [N6: nat,X4: uint32] : ( if_uint32 @ ( ord_less_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftr @ X4 @ ( code_integer_of_nat @ N6 ) ) @ zero_zero_uint32 ) ) ) ).

% shiftr_uint32_code
thf(fact_9084_nat__of__integer__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
      = N ) ).

% nat_of_integer_integer_of_nat
thf(fact_9085_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
      = ( semiri1314217659103216013at_int @ X ) ) ).

% integer_of_nat.rep_eq
thf(fact_9086_int__of__integer__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% int_of_integer_integer_of_nat
thf(fact_9087_integer__of__nat__0,axiom,
    ( ( code_integer_of_nat @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% integer_of_nat_0
thf(fact_9088_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_9089_integer__of__nat__less__0__conv,axiom,
    ! [N: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( code_integer_of_nat @ N ) @ zero_z3403309356797280102nteger ) ).

% integer_of_nat_less_0_conv
thf(fact_9090_integer__of__nat__eq__of__nat,axiom,
    code_integer_of_nat = semiri4939895301339042750nteger ).

% integer_of_nat_eq_of_nat
thf(fact_9091_integer__of__nat_Oabs__eq,axiom,
    ( code_integer_of_nat
    = ( ^ [X4: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X4 ) ) ) ) ).

% integer_of_nat.abs_eq
thf(fact_9092_integer__of__nat__numeral,axiom,
    ! [N: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% integer_of_nat_numeral
thf(fact_9093_test__bit__uint32__code,axiom,
    ( bit_se5367290876889521763uint32
    = ( ^ [X4: uint32,N6: nat] :
          ( ( ord_less_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) )
          & ( uint32_test_bit @ X4 @ ( code_integer_of_nat @ N6 ) ) ) ) ) ).

% test_bit_uint32_code
thf(fact_9094_set__bit__uint32__code,axiom,
    ( generi1993664874377053279uint32
    = ( ^ [X4: uint32,N6: nat,B7: $o] : ( if_uint32 @ ( ord_less_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_set_bit @ X4 @ ( code_integer_of_nat @ N6 ) @ B7 ) @ X4 ) ) ) ).

% set_bit_uint32_code
thf(fact_9095_shiftl__uint32__code,axiom,
    ( bit_se5742574853984576102uint32
    = ( ^ [N6: nat,X4: uint32] : ( if_uint32 @ ( ord_less_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftl @ X4 @ ( code_integer_of_nat @ N6 ) ) @ zero_zero_uint32 ) ) ) ).

% shiftl_uint32_code
thf(fact_9096_sshiftr__uint32__code,axiom,
    ( signed489701013188660438uint32
    = ( ^ [N6: nat,X4: uint32] : ( if_uint32 @ ( ord_less_nat @ N6 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X4 @ ( code_integer_of_nat @ N6 ) ) @ ( if_uint32 @ ( bit_se5367290876889521763uint32 @ X4 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ zero_zero_uint32 ) ) ) ) ).

% sshiftr_uint32_code
thf(fact_9097_uint32__sshiftr__def,axiom,
    ( uint32_sshiftr
    = ( ^ [X4: uint32,N6: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N6 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N6 ) )
          @ ( undefi7330133036835070352uint32 @ signed489701013188660438uint32 @ N6 @ X4 )
          @ ( signed489701013188660438uint32 @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).

% uint32_sshiftr_def
thf(fact_9098_uint32__sshiftr__code,axiom,
    ! [N: code_integer,W2: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_sshiftr @ W2 @ N ) )
          = ( rep_uint32 @ ( undefi7330133036835070352uint32 @ signed489701013188660438uint32 @ N @ W2 ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint32 @ ( uint32_sshiftr @ W2 @ N ) )
          = ( signed5000768011106662067l_num1 @ ( code_nat_of_integer @ N ) @ ( rep_uint32 @ W2 ) ) ) ) ) ).

% uint32_sshiftr_code

% Helper facts (45)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( if_uint32 @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( if_uint32 @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_I_Eo_J_J_T,axiom,
    ! [X: option_set_o,Y: option_set_o] :
      ( ( if_option_set_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_I_Eo_J_J_T,axiom,
    ! [X: option_set_o,Y: option_set_o] :
      ( ( if_option_set_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J_T,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ( if_option_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J_T,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ( if_option_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X: produc827990862158126777uint32,Y: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X: produc827990862158126777uint32,Y: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (6)
thf(conj_0,hypothesis,
    ( s
    = ( vEBT_set_vebt @ t ) ) ).

thf(conj_1,hypothesis,
    vEBT_invar_vebt @ t @ n ).

thf(conj_2,hypothesis,
    rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ t @ x ) @ r ) @ ( produc7507926704131184380et_nat @ a @ b ) ).

thf(conj_3,hypothesis,
    member_nat @ xa @ ( vEBT_set_vebt @ t ) ).

thf(conj_4,hypothesis,
    ~ ( member_nat @ xa @ ( vEBT_set_vebt @ ( vEBT_vebt_delete @ t @ x ) ) ) ).

thf(conj_5,conjecture,
    xa = x ).

%------------------------------------------------------------------------------